Exponential and Logarithmic Equations – Pre-Calculus (2023)

Table of Contents
Learning Objectives Using Like Bases to Solve Exponential Equations Using the One-to-One Property of Exponential Functions to Solve Exponential Equations How To Solving an Exponential Equation with a Common Base Try It How To Solving Equations by Rewriting Them to Have a Common Base Try It Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base Try It Solving an Equation with Positive and Negative Powers Try It Solving Exponential Equations Using Logarithms How To Solving an Equation Containing Powers of Different Bases Try It How To Solve an Equation of the Form y = Aekt Try It Solving an Equation That Can Be Simplified to the Form y = Aekt Try It Solving Exponential Functions in Quadratic Form Try It Using the Definition of a Logarithm to Solve Logarithmic Equations Using the Definition of a Logarithm to Solve Logarithmic Equations Using Algebra to Solve a Logarithmic Equation Try It Using Algebra Before and After Using the Definition of the Natural Logarithm Try It Using a Graph to Understand the Solution to a Logarithmic Equation Try It Using the One-to-One Property of Logarithms to Solve Logarithmic Equations Using the One-to-One Property of Logarithms to Solve Logarithmic Equations How To Solving an Equation Using the One-to-One Property of Logarithms Try It Solving Applied Problems Using Exponential and Logarithmic Equations Using the Formula for Radioactive Decay to Find the Quantity of a Substance Try It Key Equations Key Concepts Section Exercises Glossary FAQs Videos

Learning Objectives

In this section, you will:

  • Use like bases to solve exponential equations.
  • Use logarithms to solve exponential equations.
  • Use the definition of a logarithm to solve logarithmic equations.
  • Use the one-to-one property of logarithms to solve logarithmic equations.
  • Solve applied problems involving exponential and logarithmic equations.
Exponential and Logarithmic Equations – Pre-Calculus (1)

In 1859, an Australian landowner named Thomas Austin released 24 rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.

Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

Using Like Bases to Solve Exponential Equations

The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers Exponential and Logarithmic Equations – Pre-Calculus (2) Exponential and Logarithmic Equations – Pre-Calculus (3) and Exponential and Logarithmic Equations – Pre-Calculus (4) where Exponential and Logarithmic Equations – Pre-Calculus (5) Exponential and Logarithmic Equations – Pre-Calculus (6) if and only if Exponential and Logarithmic Equations – Pre-Calculus (7)

In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.

For example, consider the equation Exponential and Logarithmic Equations – Pre-Calculus (8) To solve for Exponential and Logarithmic Equations – Pre-Calculus (9) we use the division property of exponents to rewrite the right side so that both sides have the common base, Exponential and Logarithmic Equations – Pre-Calculus (10) Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for Exponential and Logarithmic Equations – Pre-Calculus (11)

Exponential and Logarithmic Equations – Pre-Calculus (12)

Using the One-to-One Property of Exponential Functions to Solve Exponential Equations

For any algebraic expressions Exponential and Logarithmic Equations – Pre-Calculus (13) and any positive real number Exponential and Logarithmic Equations – Pre-Calculus (14)

Exponential and Logarithmic Equations – Pre-Calculus (15)

How To

Given an exponential equation with the form Exponential and Logarithmic Equations – Pre-Calculus (16) where Exponential and Logarithmic Equations – Pre-Calculus (17) and Exponential and Logarithmic Equations – Pre-Calculus (18) are algebraic expressions with an unknown, solve for the unknown.

  1. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form Exponential and Logarithmic Equations – Pre-Calculus (19)
  2. Use the one-to-one property to set the exponents equal.
  3. Solve the resulting equation, Exponential and Logarithmic Equations – Pre-Calculus (20) for the unknown.

Solving an Exponential Equation with a Common Base

Solve Exponential and Logarithmic Equations – Pre-Calculus (21)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (22)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (23)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (24)

Rewriting Equations So All Powers Have the Same Base

Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.

For example, consider the equation Exponential and Logarithmic Equations – Pre-Calculus (25) We can rewrite both sides of this equation as a power of Exponential and Logarithmic Equations – Pre-Calculus (26) Then we apply the rules of exponents, along with the one-to-one property, to solve for Exponential and Logarithmic Equations – Pre-Calculus (27)

Exponential and Logarithmic Equations – Pre-Calculus (28)

How To

Given an exponential equation with unlike bases, use the one-to-one property to solve it.

  1. Rewrite each side in the equation as a power with a common base.
  2. Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form Exponential and Logarithmic Equations – Pre-Calculus (29)
  3. Use the one-to-one property to set the exponents equal.
  4. Solve the resulting equation, Exponential and Logarithmic Equations – Pre-Calculus (30) for the unknown.

Solving Equations by Rewriting Them to Have a Common Base

Solve Exponential and Logarithmic Equations – Pre-Calculus (31)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (32)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (33)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (34)

Solving Equations by Rewriting Roots with Fractional Exponents to Have a Common Base

Solve Exponential and Logarithmic Equations – Pre-Calculus (35)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (36)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (37)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (38)

Do all exponential equations have a solution? If not, how can we tell if there is a solution during the problem-solving process?

No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.

Solving an Equation with Positive and Negative Powers

Solve Exponential and Logarithmic Equations – Pre-Calculus (39)

Show Solution

This equation has no solution. There is no real value of Exponential and Logarithmic Equations – Pre-Calculus (40) that will make the equation a true statement because any power of a positive number is positive.

Analysis

(Figure) shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.

Exponential and Logarithmic Equations – Pre-Calculus (41)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (42)

Show Solution

The equation has no solution.

Solving Exponential Equations Using Logarithms

Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since Exponential and Logarithmic Equations – Pre-Calculus (43) is equivalent to Exponential and Logarithmic Equations – Pre-Calculus (44) we may apply logarithms with the same base on both sides of an exponential equation.

How To

Given an exponential equation in which a common base cannot be found, solve for the unknown.

  1. Apply the logarithm of both sides of the equation.
    • If one of the terms in the equation has base 10, use the common logarithm.
    • If none of the terms in the equation has base 10, use the natural logarithm.
  2. Use the rules of logarithms to solve for the unknown.

Solving an Equation Containing Powers of Different Bases

Solve Exponential and Logarithmic Equations – Pre-Calculus (45)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (46)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (47)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (48)

Is there any way to solve Exponential and Logarithmic Equations – Pre-Calculus (49)

Yes. The solution is Exponential and Logarithmic Equations – Pre-Calculus (50)

Equations Containing e

One common type of exponential equations are those with base Exponential and Logarithmic Equations – Pre-Calculus (51) This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base Exponential and Logarithmic Equations – Pre-Calculus (52) on either side, we can use the natural logarithm to solve it.

How To

Given an equation of the form Exponential and Logarithmic Equations – Pre-Calculus (53) solve for Exponential and Logarithmic Equations – Pre-Calculus (54)

  1. Divide both sides of the equation by Exponential and Logarithmic Equations – Pre-Calculus (55)
  2. Apply the natural logarithm of both sides of the equation.
  3. Divide both sides of the equation by Exponential and Logarithmic Equations – Pre-Calculus (56)

Solve an Equation of the Form y = Aekt

Solve Exponential and Logarithmic Equations – Pre-Calculus (57)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (58)

Analysis

Using laws of logs, we can also write this answer in the form Exponential and Logarithmic Equations – Pre-Calculus (59) If we want a decimal approximation of the answer, we use a calculator.

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (60)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (61) or Exponential and Logarithmic Equations – Pre-Calculus (62)

(Video) Pre-Calculus: Solving Exponential and Logarithmic Equations

Does every equation of the form Exponential and Logarithmic Equations – Pre-Calculus (63) have a solution?

No. There is a solution when Exponential and Logarithmic Equations – Pre-Calculus (64) and when Exponential and Logarithmic Equations – Pre-Calculus (65) and Exponential and Logarithmic Equations – Pre-Calculus (66) are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is Exponential and Logarithmic Equations – Pre-Calculus (67)

Solving an Equation That Can Be Simplified to the Form y = Aekt

Solve Exponential and Logarithmic Equations – Pre-Calculus (68)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (69)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (70)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (71)

Extraneous Solutions

Sometimes the methods used to solve an equation introduce an extraneous solution, which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.

Solving Exponential Functions in Quadratic Form

Solve Exponential and Logarithmic Equations – Pre-Calculus (72)

Analysis

When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation Exponential and Logarithmic Equations – Pre-Calculus (74) because a positive number never equals a negative number. The solution Exponential and Logarithmic Equations – Pre-Calculus (75) is not a real number, and in the real number system this solution is rejected as an extraneous solution.

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (76)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (77)

Does every logarithmic equation have a solution?

No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.

Using the Definition of a Logarithm to Solve Logarithmic Equations

We have already seen that every logarithmic equation Exponential and Logarithmic Equations – Pre-Calculus (78) is equivalent to the exponential equation Exponential and Logarithmic Equations – Pre-Calculus (79) We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.

For example, consider the equation Exponential and Logarithmic Equations – Pre-Calculus (80) To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for Exponential and Logarithmic Equations – Pre-Calculus (81)

Exponential and Logarithmic Equations – Pre-Calculus (82)

Using the Definition of a Logarithm to Solve Logarithmic Equations

For any algebraic expression Exponential and Logarithmic Equations – Pre-Calculus (83) and real numbers Exponential and Logarithmic Equations – Pre-Calculus (84) and Exponential and Logarithmic Equations – Pre-Calculus (85) where Exponential and Logarithmic Equations – Pre-Calculus (86)

Exponential and Logarithmic Equations – Pre-Calculus (87)

Using Algebra to Solve a Logarithmic Equation

Solve Exponential and Logarithmic Equations – Pre-Calculus (88)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (89)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (90)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (91)

Using Algebra Before and After Using the Definition of the Natural Logarithm

Solve Exponential and Logarithmic Equations – Pre-Calculus (92)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (93)

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (94)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (95)

Using a Graph to Understand the Solution to a Logarithmic Equation

Solve Exponential and Logarithmic Equations – Pre-Calculus (96)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (97)

(Figure) represents the graph of the equation. On the graph, the x-coordinate of the point at which the two graphs intersect is close to 20. In other words Exponential and Logarithmic Equations – Pre-Calculus (98) A calculator gives a better approximation: Exponential and Logarithmic Equations – Pre-Calculus (99)

Exponential and Logarithmic Equations – Pre-Calculus (100)

Try It

Use a graphing calculator to estimate the approximate solution to the logarithmic equation Exponential and Logarithmic Equations – Pre-Calculus (104) to 2 decimal places.

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (105)

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers Exponential and Logarithmic Equations – Pre-Calculus (106) Exponential and Logarithmic Equations – Pre-Calculus (107) Exponential and Logarithmic Equations – Pre-Calculus (108) and any positive real number Exponential and Logarithmic Equations – Pre-Calculus (109) where Exponential and Logarithmic Equations – Pre-Calculus (110)

Exponential and Logarithmic Equations – Pre-Calculus (111)

For example,

Exponential and Logarithmic Equations – Pre-Calculus (112)

So, if Exponential and Logarithmic Equations – Pre-Calculus (113) then we can solve for Exponential and Logarithmic Equations – Pre-Calculus (114) and we get Exponential and Logarithmic Equations – Pre-Calculus (115) To check, we can substitute Exponential and Logarithmic Equations – Pre-Calculus (116) into the original equation: Exponential and Logarithmic Equations – Pre-Calculus (117) In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.

For example, consider the equation Exponential and Logarithmic Equations – Pre-Calculus (118) To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for Exponential and Logarithmic Equations – Pre-Calculus (119)

Exponential and Logarithmic Equations – Pre-Calculus (120)

To check the result, substitute Exponential and Logarithmic Equations – Pre-Calculus (121) into Exponential and Logarithmic Equations – Pre-Calculus (122)

Exponential and Logarithmic Equations – Pre-Calculus (123)

Using the One-to-One Property of Logarithms to Solve Logarithmic Equations

For any algebraic expressions Exponential and Logarithmic Equations – Pre-Calculus (124) and Exponential and Logarithmic Equations – Pre-Calculus (125) and any positive real number Exponential and Logarithmic Equations – Pre-Calculus (126) where Exponential and Logarithmic Equations – Pre-Calculus (127)

Exponential and Logarithmic Equations – Pre-Calculus (128)

Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.

How To

Given an equation containing logarithms, solve it using the one-to-one property.

  1. Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form Exponential and Logarithmic Equations – Pre-Calculus (129)
  2. Use the one-to-one property to set the arguments equal.
  3. Solve the resulting equation, Exponential and Logarithmic Equations – Pre-Calculus (130) for the unknown.

Solving an Equation Using the One-to-One Property of Logarithms

Solve Exponential and Logarithmic Equations – Pre-Calculus (131)

(Video) Pre-Calculus 3.4: Exponential and Logarithmic Equations part 1

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (132)

Analysis

There are two solutions: Exponential and Logarithmic Equations – Pre-Calculus (133) or Exponential and Logarithmic Equations – Pre-Calculus (134) The solution Exponential and Logarithmic Equations – Pre-Calculus (135) is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.

Try It

Solve Exponential and Logarithmic Equations – Pre-Calculus (136)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (137) or Exponential and Logarithmic Equations – Pre-Calculus (138)

Solving Applied Problems Using Exponential and Logarithmic Equations

In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.

One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life. (Figure) lists the half-life for several of the more common radioactive substances.

SubstanceUseHalf-life
gallium-67nuclear medicine80 hours
cobalt-60manufacturing5.3 years
technetium-99mnuclear medicine6 hours
americium-241construction432 years
carbon-14archeological dating5,715 years
uranium-235atomic power703,800,000 years

We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:

Exponential and Logarithmic Equations – Pre-Calculus (139)

where

  • Exponential and Logarithmic Equations – Pre-Calculus (140) is the amount initially present
  • Exponential and Logarithmic Equations – Pre-Calculus (141) is the half-life of the substance
  • Exponential and Logarithmic Equations – Pre-Calculus (142) is the time period over which the substance is studied
  • Exponential and Logarithmic Equations – Pre-Calculus (143) is the amount of the substance present after time Exponential and Logarithmic Equations – Pre-Calculus (144)

Using the Formula for Radioactive Decay to Find the Quantity of a Substance

How long will it take for ten percent of a 1000-gram sample of uranium-235 to decay?

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (145)

Analysis

Ten percent of 1000 grams is 100 grams. If 100 grams decay, the amount of uranium-235 remaining is 900 grams.

Try It

How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (146)

Access these online resources for additional instruction and practice with exponential and logarithmic equations.

Key Equations

One-to-one property for exponential functionsFor any algebraic expressions Exponential and Logarithmic Equations – Pre-Calculus (147) and Exponential and Logarithmic Equations – Pre-Calculus (148) and any positive real number Exponential and Logarithmic Equations – Pre-Calculus (149) where< Exponential and Logarithmic Equations – Pre-Calculus (150) if and only if Exponential and Logarithmic Equations – Pre-Calculus (151) </td>
Definition of a logarithmFor any algebraic expression S and positive real numbers Exponential and Logarithmic Equations – Pre-Calculus (152) and Exponential and Logarithmic Equations – Pre-Calculus (153) where Exponential and Logarithmic Equations – Pre-Calculus (154) < Exponential and Logarithmic Equations – Pre-Calculus (155) if and only if Exponential and Logarithmic Equations – Pre-Calculus (156) </td>
One-to-one property for logarithmic functionsFor any algebraic expressions S and T and any positive real number Exponential and Logarithmic Equations – Pre-Calculus (157) where Exponential and Logarithmic Equations – Pre-Calculus (158) < Exponential and Logarithmic Equations – Pre-Calculus (159) if and only if Exponential and Logarithmic Equations – Pre-Calculus (160) </td>

Key Concepts

  • We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
  • When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See (Figure).
  • When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See (Figure), (Figure), and (Figure).
  • When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See (Figure).
  • We can solve exponential equations with base Exponential and Logarithmic Equations – Pre-Calculus (161) by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See (Figure) and (Figure).
  • After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See (Figure).
  • When given an equation of the form Exponential and Logarithmic Equations – Pre-Calculus (162) where Exponential and Logarithmic Equations – Pre-Calculus (163) is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation Exponential and Logarithmic Equations – Pre-Calculus (164) and solve for the unknown. See (Figure) and (Figure).
  • We can also use graphing to solve equations with the form Exponential and Logarithmic Equations – Pre-Calculus (165) We graph both equations Exponential and Logarithmic Equations – Pre-Calculus (166) and Exponential and Logarithmic Equations – Pre-Calculus (167) on the same coordinate plane and identify the solution as the x-value of the intersecting point. See (Figure).
  • When given an equation of the form Exponential and Logarithmic Equations – Pre-Calculus (168) where Exponential and Logarithmic Equations – Pre-Calculus (169) and Exponential and Logarithmic Equations – Pre-Calculus (170) are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation Exponential and Logarithmic Equations – Pre-Calculus (171) for the unknown. See (Figure).
  • Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See (Figure).

Section Exercises

Verbal

1. How can an exponential equation be solved?

Show Solution

Determine first if the equation can be rewritten so that each side uses the same base. If so, the exponents can be set equal to each other. If the equation cannot be rewritten so that each side uses the same base, then apply the logarithm to each side and use properties of logarithms to solve.

2. When does an extraneous solution occur? How can an extraneous solution be recognized?

3. When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

Show Solution

The one-to-one property can be used if both sides of the equation can be rewritten as a single logarithm with the same base. If so, the arguments can be set equal to each other, and the resulting equation can be solved algebraically. The one-to-one property cannot be used when each side of the equation cannot be rewritten as a single logarithm with the same base.

Algebraic

For the following exercises, use like bases to solve the exponential equation.

4. Exponential and Logarithmic Equations – Pre-Calculus (172)

6. Exponential and Logarithmic Equations – Pre-Calculus (175)

7. Exponential and Logarithmic Equations – Pre-Calculus (176)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (177)

8. Exponential and Logarithmic Equations – Pre-Calculus (178)

9. Exponential and Logarithmic Equations – Pre-Calculus (179)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (180)

10. Exponential and Logarithmic Equations – Pre-Calculus (181)

For the following exercises, use logarithms to solve.

11. Exponential and Logarithmic Equations – Pre-Calculus (182)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (183)

12. Exponential and Logarithmic Equations – Pre-Calculus (184)

13. Exponential and Logarithmic Equations – Pre-Calculus (185)

Show Solution

No solution

14. Exponential and Logarithmic Equations – Pre-Calculus (186)

15. Exponential and Logarithmic Equations – Pre-Calculus (187)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (188)

16. Exponential and Logarithmic Equations – Pre-Calculus (189)

17. Exponential and Logarithmic Equations – Pre-Calculus (190)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (191)

18. Exponential and Logarithmic Equations – Pre-Calculus (192)

19. Exponential and Logarithmic Equations – Pre-Calculus (193)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (194)

20. Exponential and Logarithmic Equations – Pre-Calculus (195)

21. Exponential and Logarithmic Equations – Pre-Calculus (196)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (197)

22. Exponential and Logarithmic Equations – Pre-Calculus (198)

(Video) Solving Logarithmic Equations with Exponentials (Precalculus - College Algebra 63)

23. Exponential and Logarithmic Equations – Pre-Calculus (199)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (200)

24. Exponential and Logarithmic Equations – Pre-Calculus (201)

25. Exponential and Logarithmic Equations – Pre-Calculus (202)

Show Solution

no solution

26. Exponential and Logarithmic Equations – Pre-Calculus (203)

27. Exponential and Logarithmic Equations – Pre-Calculus (204)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (205)

28. Exponential and Logarithmic Equations – Pre-Calculus (206)

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.

29. Exponential and Logarithmic Equations – Pre-Calculus (207)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (208)

30. Exponential and Logarithmic Equations – Pre-Calculus (209)

For the following exercises, use the definition of a logarithm to solve the equation.

31. Exponential and Logarithmic Equations – Pre-Calculus (210)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (211)

32. Exponential and Logarithmic Equations – Pre-Calculus (212)

33. Exponential and Logarithmic Equations – Pre-Calculus (213)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (214)

34. Exponential and Logarithmic Equations – Pre-Calculus (215)

35. Exponential and Logarithmic Equations – Pre-Calculus (216)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (217)

For the following exercises, use the one-to-one property of logarithms to solve.

36. Exponential and Logarithmic Equations – Pre-Calculus (218)

37. Exponential and Logarithmic Equations – Pre-Calculus (219)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (220)

38. Exponential and Logarithmic Equations – Pre-Calculus (221)

39. Exponential and Logarithmic Equations – Pre-Calculus (222)

Show Solution

No solution

40. Exponential and Logarithmic Equations – Pre-Calculus (223)

41. Exponential and Logarithmic Equations – Pre-Calculus (224)

Show Solution

No solution

42. Exponential and Logarithmic Equations – Pre-Calculus (225)

43. Exponential and Logarithmic Equations – Pre-Calculus (226)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (227)

For the following exercises, solve each equation for Exponential and Logarithmic Equations – Pre-Calculus (228)

44. Exponential and Logarithmic Equations – Pre-Calculus (229)

46. Exponential and Logarithmic Equations – Pre-Calculus (232)

47. Exponential and Logarithmic Equations – Pre-Calculus (233)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (234)

48. Exponential and Logarithmic Equations – Pre-Calculus (235)

49. Exponential and Logarithmic Equations – Pre-Calculus (236)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (237)

50. Exponential and Logarithmic Equations – Pre-Calculus (238)

Graphical

For the following exercises, solve the equation for Exponential and Logarithmic Equations – Pre-Calculus (239) if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution.

51. Exponential and Logarithmic Equations – Pre-Calculus (240)

Show Solution

Exponential and Logarithmic Equations – Pre-Calculus (241)

Exponential and Logarithmic Equations – Pre-Calculus (242)

52. Exponential and Logarithmic Equations – Pre-Calculus (243)

(Video) Solving Exponential and Logarithmic Equations | Pre-Calculus Review

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For the following exercises, solve for the indicated value, and graph the situation showing the solution point.

65. An account with an initial deposit of Exponential and Logarithmic Equations – Pre-Calculus (267) earns Exponential and Logarithmic Equations – Pre-Calculus (268) annual interest, compounded continuously. How much will the account be worth after 20 years?

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about Exponential and Logarithmic Equations – Pre-Calculus (269)

Exponential and Logarithmic Equations – Pre-Calculus (270)

66. The formula for measuring sound intensity in decibels Exponential and Logarithmic Equations – Pre-Calculus (271) is defined by the equation Exponential and Logarithmic Equations – Pre-Calculus (272) where Exponential and Logarithmic Equations – Pre-Calculus (273) is the intensity of the sound in watts per square meter and Exponential and Logarithmic Equations – Pre-Calculus (274) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of Exponential and Logarithmic Equations – Pre-Calculus (275) watts per square meter?

67. The population of a small town is modeled by the equation Exponential and Logarithmic Equations – Pre-Calculus (276) where Exponential and Logarithmic Equations – Pre-Calculus (277) is measured in years. In approximately how many years will the town’s population reach Exponential and Logarithmic Equations – Pre-Calculus (278)

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about 5 years

Exponential and Logarithmic Equations – Pre-Calculus (279)

Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate Exponential and Logarithmic Equations – Pre-Calculus (280) to 3 decimal places.

68. Exponential and Logarithmic Equations – Pre-Calculus (281) using the common log.

69. Exponential and Logarithmic Equations – Pre-Calculus (282) using the natural log

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70. Exponential and Logarithmic Equations – Pre-Calculus (284) using the common log

71. Exponential and Logarithmic Equations – Pre-Calculus (285) using the common log

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72. Exponential and Logarithmic Equations – Pre-Calculus (287) using the natural log

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

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76. Atmospheric pressure Exponential and Logarithmic Equations – Pre-Calculus (293) in pounds per square inch is represented by the formula Exponential and Logarithmic Equations – Pre-Calculus (294) where Exponential and Logarithmic Equations – Pre-Calculus (295) is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of Exponential and Logarithmic Equations – Pre-Calculus (296) pounds per square inch? (Hint: there are 5280 feet in a mile)

77. The magnitude M of an earthquake is represented by the equation Exponential and Logarithmic Equations – Pre-Calculus (297) where Exponential and Logarithmic Equations – Pre-Calculus (298) is the amount of energy released by the earthquake in joules and Exponential and Logarithmic Equations – Pre-Calculus (299) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing Exponential and Logarithmic Equations – Pre-Calculus (300) joules of energy?

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about Exponential and Logarithmic Equations – Pre-Calculus (301)

Extensions

78. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that Exponential and Logarithmic Equations – Pre-Calculus (302)

79. Recall the formula for continually compounding interest, Exponential and Logarithmic Equations – Pre-Calculus (303) Use the definition of a logarithm along with properties of logarithms to solve the formula for time Exponential and Logarithmic Equations – Pre-Calculus (304) such that Exponential and Logarithmic Equations – Pre-Calculus (305) is equal to a single logarithm.

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Exponential and Logarithmic Equations – Pre-Calculus (306)

80. Recall the compound interest formula Exponential and Logarithmic Equations – Pre-Calculus (307) Use the definition of a logarithm along with properties of logarithms to solve the formula for time Exponential and Logarithmic Equations – Pre-Calculus (308)

81. Newton’s Law of Cooling states that the temperature Exponential and Logarithmic Equations – Pre-Calculus (309) of an object at any time t can be described by the equation Exponential and Logarithmic Equations – Pre-Calculus (310) where Exponential and Logarithmic Equations – Pre-Calculus (311) is the temperature of the surrounding environment, Exponential and Logarithmic Equations – Pre-Calculus (312) is the initial temperature of the object, and Exponential and Logarithmic Equations – Pre-Calculus (313) is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time Exponential and Logarithmic Equations – Pre-Calculus (314) such that Exponential and Logarithmic Equations – Pre-Calculus (315) is equal to a single logarithm.

(Video) Solving Exponential Equations with Logarithms (Precalculus - College Algebra 64)

Glossary

extraneous solution
a solution introduced while solving an equation that does not satisfy the conditions of the original equation

FAQs

What is exponential and logarithmic equations? ›

An exponential equation is an equation in which the variable appears in an exponent. A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first see whether you can write both sides of the equation as powers of the same number.

What is logarithm in precalculus? ›

When the base a is equal to e, the logarithm has a special name: the natural logarithm, which we write as ln x. This natural logarithmic function is the inverse of the exponential . Thus, This means that the following two equations must both be true.

What are the 3 types of logarithms? ›

The most common types of logarithms are common logarithms, where the base is 10, binary logarithms, where the base is 2, and natural logarithms, where the base is e ≈ 2.71828.

What is the difference between exponential and logarithmic functions? ›

An exponential function has the form ax, where a is a constant; examples are 2x, 10x, ex. The logarithmic functions are the inverses of the exponential functions, that is, functions that "undo'' the exponential functions, just as, for example, the cube root function "undoes'' the cube function: 3√23=2.

What are the 3 exponential rules? ›

The first law states that to multiply two exponential functions with the same base, we simply add the exponents. The second law states that to divide two exponential functions with the same base, we subtract the exponents. The third law states that in order to raise a power to a new power, we multiply the exponents.

What is the exponential rule in calculus? ›

The exponential rule is a special case of the chain rule. It is useful when finding the derivative of e raised to the power of a function. The exponential rule states that this derivative is e to the power of the function times the derivative of the function.

Which is an example of logarithmic equation? ›

Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8. In the same fashion, since 102 = 100, then 2 = log10 100.

How do you find the logarithmic equation? ›

The logarithmic function for x = 2y is written as y = log2 x or f(x) = log2 x. The number 2 is still called the base. In general, y = logb x is read, “y equals log to the base b of x,” or more simply, “y equals log base b of x.” As with exponential functions, b > 0 and b ≠ 1.
...
x = 3yy
−1
10
31
92
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What is the difference between logarithmic and exponential growth? ›

The logarithm is the mathematical inverse of the exponential, so while exponential growth starts slowly and then speeds up faster and faster, logarithm growth starts fast and then gets slower and slower.

What are the two types of logarithms? ›

Two kinds of logarithms are often used in chemistry: common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number.

Why is it called logarithmic? ›

Napier coined the term for logarithm in Middle Latin, “logarithmus,” derived from the Greek, literally meaning, “ratio-number,” from logos “proportion, ratio, word” + arithmos “number”. The common logarithm of a number is the index of that power of ten which equals the number.

Are logarithms calculus or algebra? ›

Logarithms are neither calculus nor algebra, they are operators. They are the answer to the question: what power do i need to raise this base to to get the resulting number? I.e.: In base 2, the logarithm of 16 is 4, or: 2 to the power of 4 = 16.

What are the 7 Laws of logarithms? ›

The names of these rules are:
  • Product rule.
  • Division rule.
  • Power rule/Exponential Rule.
  • Change of base rule.
  • Base switch rule.
  • Derivative of log.
  • Integral of log.

What are the rules of logarithm? ›

The rules apply for any logarithm logbx, except that you have to replace any occurence of e with the new base b. The natural log was defined by equations (1) and (2).
...
Basic rules for logarithms.
Rule or special caseFormula
Quotientln(x/y)=ln(x)−ln(y)
Log of powerln(xy)=yln(x)
Log of eln(e)=1
Log of oneln(1)=0
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What is the logarithmic function called? ›

the natural logarithm

What are the importance of exponential and logarithmic functions? ›

The exponential and the logarithmic functions are perhaps the most important functions you'll encounter whenever dealing with a physical problem. They are the inverse of each other and can be used to represent a large range of numbers very conveniently.

What is the similarities between exponential and logarithmic functions? ›

Expert-Verified Answer

Logarithmic functions are the inverse of exponential functions which means y = loga x, where "a" is greater than zero and not equal to one.

Why are exponential and logarithmic functions important? ›

These functions are used to study many naturally occurring phenomena such as population growth, exponential decay of radioactive matter, and growth of investments. Addition, subtraction, multiplication, and division can be used to create a new function from two or more functions.

What are the 4 types of exponents? ›

Exponents can be observed in 4 different types namely, positive, negative, zero and rational/fractional. The number's value can be interpreted by using the exponent as the total number of times the base number has to be multiplied with the same base.

What are the 5 laws of exponent? ›

In Mathematics, there are different laws of exponents.
...
The different Laws of exponents are:
  • am×an = a. m+n
  • am/an = a. m-n
  • (am)n = a. mn
  • an/bn = (a/b) n
  • a0 = 1.
  • a-m = 1/a. m

What is power 3 called? ›

In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together.

What are the two types of exponential equations? ›

There are two types of exponential functions: exponential growth and exponential decay. In the function f (x) = bx when b > 1, the function represents exponential growth. In the function f (x) = bx when 0 < b < 1, the function represents exponential decay.

How do you identify an exponential equation? ›

An exponential function is a function of the form f(x)=ab^x for positive real numbers a and b.

How do you know if an equation is exponential? ›

In an exponential function, the independent variable, or x-value, is the exponent, while the base is a constant. For example, y = 2x would be an exponential function. Here's what that looks like. The formula for an exponential function is y = abx, where a and b are constants.

What is the main function of exponential? ›

Exponential functions frequently arise and quantitatively describe a number of phenomena in physics, such as radioactive decay, in which the rate of change in a process or substance depends directly on its current value.

What is a exponential function example? ›

An example of an exponential function is the growth of bacteria. Some bacteria double every hour. If you start with 1 bacterium and it doubles every hour, you will have 2x bacteria after x hours. This can be written as f(x) = 2x.

What are the main points of an exponential function? ›

The graphs of all exponential functions have these characteristics. They all contain the point (0, 1), because a0 = 1. The x-axis is always an asymptote.

What is the first step to solving logarithmic equations? ›

Step 1: The first step in solving a logarithmic equation is to isolate the logarithmic term on one side of the equation. Our equation log 7 (x – 3) = 17 is already in this form so we can move on to the next step.

What are the 4 properties of logarithm? ›

The Four Basic Properties of Logs

logb(xy) = logbx + logby. logb(x/y) = logbx - logby. logb(xn) = n logbx. logbx = logax / logab.

What are the four formulas for logarithms? ›

Basic Logarithm Formulas
  • ⁡ ( x y ) = log b ⁡ ( x ) + log b ⁡
  • log b ⁡ ( x y ) = log b ⁡ ( x ) – log b ⁡
  • log b ⁡ ( x d ) = d log b ⁡
  • c log b ⁡ ( x ) + d log b ⁡ ( y ) = log b ⁡
  • ⁡ ( a + c ) = log b ⁡ a + log b ⁡
  • log b ⁡ ( a − c ) = log b ⁡ a + log b ⁡

Why is logarithmic important? ›

Logarithmic functions are important largely because of their relationship to exponential functions. Logarithms can be used to solve exponential equations and to explore the properties of exponential functions.

What is logarithmic equation with example? ›

LOGARITHMIC EQUATIONS
DefinitionAny equation in the variable x that contains a logarithm is called a logarithmic equation.
Recall the definition of a logarithm. This definition will be important to understand in order to be able to solve logarithmic equations.
ExamplesEXAMPLES OF LOGARITHMIC EQUATIONS
Log2 x = -5
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What is logarithm explain? ›

logarithm, the exponent or power to which a base must be raised to yield a given number. Expressed mathematically, x is the logarithm of n to the base b if bx = n, in which case one writes x = logb n. For example, 23 = 8; therefore, 3 is the logarithm of 8 to base 2, or 3 = log2 8.

What is a logarithmic equation used for? ›

Applications of logarithmic functions include the pH scale in chemistry, sound intensity, the Richter scale for earthquakes, and Newton's law of cooling.

What are the two types of logarithmic equations? ›

Math Skills - Logarithms. Two kinds of logarithms are often used in chemistry: common (or Briggian) logarithms and natural (or Napierian) logarithms. The power to which a base of 10 must be raised to obtain a number is called the common logarithm (log) of the number.

What are the 7 rules of logarithms? ›

The names of these rules are:
  • Product rule.
  • Division rule.
  • Power rule/Exponential Rule.
  • Change of base rule.
  • Base switch rule.
  • Derivative of log.
  • Integral of log.

Why is logarithm so important? ›

Logarithmic functions are important largely because of their relationship to exponential functions. Logarithms can be used to solve exponential equations and to explore the properties of exponential functions.

What is the function of exponential? ›

An exponential function is a Mathematical function in the form f (x) = ax, where “x” is a variable and “a” is a constant which is called the base of the function and it should be greater than 0. The most commonly used exponential function base is the transcendental number e, which is approximately equal to 2.71828.

What is a real life example of a logarithmic function? ›

Using Logarithmic Functions

Some examples of this include sound (decibel measures), earthquakes (Richter scale), the brightness of stars, and chemistry (pH balance, a measure of acidity and alkalinity).

What are the types of logarithmic functions? ›

The logarithmic functions are broadly classified into two types, based on the base of the logarithms. We have natural logarithms and common logarithms. Natural logarithms are logarithms to the base 'e', and common logarithms are logarithms to the base of 10.

Videos

1. Precalculus: Exponential and Logarithmic Equations (Video #25)
(Math TV with Professor V)
2. Introduction to Solving Logarithms and Exponentials (Precalculus - College Algebra 57)
(Professor Leonard)
3. Logarithms Review - Exponential Form - Graphing Functions & Solving Equations - Algebra
(The Organic Chemistry Tutor)
4. Solving Logarithmic Equations
(The Organic Chemistry Tutor)
5. Pre-Calc: 3.4 Exponential and Logarithmic Equations
(Joe Schiavone)
6. Pre-Calculus 3.4: Exponential and Logarithmic Equations part 2
(Justin Backeberg)
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