# Exponential and Logarithmic Functions

## Exponential and Logarithmic Equations


### Learning Objectives

1. Solve Exponential Equations. (IA 10.2.2)
2. Solve Logarithmic Equations. (IA 10.3.4)

### Objective 1: Solve Exponential Equations. (IA 10.2.2)


Equations that include an exponential expression  are called exponential equations. There are two types of exponential equations: those with the common base on each side, and those without a common base.

Type 1: Possible common base on each side: Use properties of exponents to rewrite each side with a common base. Use base-exponent property to set exponents equal to each other and solve for x.
Type 2: No possible common base: Use properties of exponents to rewrite each side in terms of one exponential expression. Take the log or ln of each side and use the power rule to bring down the power. Solve the remaining equation for x.


### Practice Makes Perfect

Solve. Find the exact answer and then approximate it to three decimal places.

### Objective 2: Solving Logarithmic Equations.  (IA 10.3.4)


There are two types of logarithmic equations: those with log terms on just one side of the equation or those with log terms on each side of the equation.  Since the domain of logarithmic functions is positive numbers only, make sure to check the solutions.
Type 1: Log terms on one side of the equation: Use properties of logs to rewrite a side with just one log term. Convert to exponential notation and solve for x.
If  then  .
Type 2: Log terms on both sides of equation: First, use log properties to rewrite each side in terms of a single log expression, if necessary. Use the one-to-one property of logarithmic equality to set arguments equal to one another. Solve the resulting equation for x.

### Practice Makes Perfect


Don’t forget to check your solutions.

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 
:

### 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 
 

### 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.

### 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.

### 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.

### 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 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 
 

### 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.  lists the half-life for several of the more common radioactive substances.
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
1.  
 is the amount initially present
2.  
 is the half-life of the substance
3.  
 is the time period over which the substance is studied
4.  
 is the amount of the substance present after time 
 

### Key Equations


### Key Concepts

1. 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.
2. 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 .
3. 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 , , and .
4. When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See .
5. 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  and .
6. After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See .
7. 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  and .
8. 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 .
9. 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 .
10. 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 .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, use like bases to solve the exponential equation.
For the following exercises, use logarithms to solve.
For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation.
For the following exercises, use the definition of a logarithm to solve the equation.
For the following exercises, use the one-to-one property of logarithms to solve.
For the following exercises, solve each equation for 
 

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

### Technology 

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places.
For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth.

### Extensions 
