# Exponential and Logarithmic Equations – Pre-Calculus (2023)

### 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.

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 and where if and only if

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 To solve for we use the division property of exponents to rewrite the right side so that both sides have the common base, Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for

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

For any algebraic expressions and any positive real number

### How To

Given an exponential equation with the form where and 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
2. Use the one-to-one property to set the exponents equal.
3. Solve the resulting equation, for the unknown.

Solve

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### Try It

Solve

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#### 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 We can rewrite both sides of this equation as a power of Then we apply the rules of exponents, along with the one-to-one property, to solve for

### 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
3. Use the one-to-one property to set the exponents equal.
4. Solve the resulting equation, for the unknown.

Solve

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Solve

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Solve

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### Try It

Solve

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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

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This equation has no solution. There is no real value of 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.

### Try It

Solve

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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 is equivalent to 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.

Solve

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### Try It

Solve

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Is there any way to solve

Yes. The solution is

#### Equations Containing e

One common type of exponential equations are those with base 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 on either side, we can use the natural logarithm to solve it.

### How To

Given an equation of the form solve for

1. Divide both sides of the equation by
2. Apply the natural logarithm of both sides of the equation.
3. Divide both sides of the equation by

### Solve an Equation of the Form y = Aekt

Solve

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#### Analysis

Using laws of logs, we can also write this answer in the form If we want a decimal approximation of the answer, we use a calculator.

### Try It

Solve

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or

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

Does every equation of the form have a solution?

No. There is a solution when and when and 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

Solve

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### Try It

Solve

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#### 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

#### 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 because a positive number never equals a negative number. The solution is not a real number, and in the real number system this solution is rejected as an extraneous solution.

### Try It

Solve

Show Solution

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 is equivalent to the exponential equation 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 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

### Using the Definition of a Logarithm to Solve Logarithmic Equations

For any algebraic expression and real numbers and where

Solve

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Solve

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Solve

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Solve

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### Using a Graph to Understand the Solution to a Logarithmic Equation

Solve

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(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 A calculator gives a better approximation:

### Try It

Use a graphing calculator to estimate the approximate solution to the logarithmic equation to 2 decimal places.

Show Solution

### 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 and any positive real number where

For example,

So, if then we can solve for and we get To check, we can substitute into the original equation: 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 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

To check the result, substitute into

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

For any algebraic expressions and and any positive real number where

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
2. Use the one-to-one property to set the arguments equal.
3. Solve the resulting equation, for the unknown.

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

Solve

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

Show Solution

#### Analysis

There are two solutions: or The solution is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.

Solve

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or

### 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:

where

• is the amount initially present
• is the half-life of the substance
• is the time period over which the substance is studied
• is the amount of the substance present after time

### 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?

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#### 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

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

### Key Equations

 One-to-one property for exponential functions For any algebraic expressions and and any positive real number where< if and only if Definition of a logarithm For any algebraic expression S and positive real numbers and where < if and only if One-to-one property for logarithmic functions For any algebraic expressions S and T and any positive real number where < if and only if

### 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 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 where is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation and solve for the unknown. See (Figure) and (Figure).
• We can also use graphing to solve equations with the form We graph both equations and 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 where and are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation 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.

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10.

For the following exercises, use logarithms to solve.

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

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no solution

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

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

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

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For the following exercises, solve each equation for

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#### Graphical

For the following exercises, solve the equation for 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.

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

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No solution

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64.

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 earns annual interest, compounded continuously. How much will the account be worth after 20 years?

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66. The formula for measuring sound intensity in decibels is defined by the equation where is the intensity of the sound in watts per square meter and 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 watts per square meter?

67. The population of a small town is modeled by the equation where is measured in years. In approximately how many years will the town’s population reach

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#### Technology

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate to 3 decimal places.

68. using the common log.

69. using the natural log

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70. using the common log

71. using the common log

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72. 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.

73.

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76. Atmospheric pressure in pounds per square inch is represented by the formula where 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 pounds per square inch? (Hint: there are 5280 feet in a mile)

77. The magnitude M of an earthquake is represented by the equation where is the amount of energy released by the earthquake in joules and is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing joules of energy?

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#### Extensions

78. Use the definition of a logarithm along with the one-to-one property of logarithms to prove that

79. Recall the formula for continually compounding interest, Use the definition of a logarithm along with properties of logarithms to solve the formula for time such that is equal to a single logarithm.

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80. Recall the compound interest formula Use the definition of a logarithm along with properties of logarithms to solve the formula for time

81. Newton’s Law of Cooling states that the temperature of an object at any time t can be described by the equation where is the temperature of the surrounding environment, is the initial temperature of the object, and is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time such that is equal to a single logarithm.

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

Show Solution

### 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
1 more row

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
2 more rows

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? ›

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)
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2. Introduction to Solving Logarithms and Exponentials (Precalculus - College Algebra 57)
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4. Solving Logarithmic Equations
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5. Pre-Calc: 3.4 Exponential and Logarithmic Equations
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6. Pre-Calculus 3.4: Exponential and Logarithmic Equations part 2
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