# Exponential and Logarithmic Functions

## Solve Exponential and Logarithmic Equations


### Solve Logarithmic Equations Using the Properties of Logarithms

In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.
If our equation has two logarithms we can use a property that says that if  then it is true that  This is the One-to-One Property of Logarithmic Equations.
To use this property, we must be certain that both sides of the equation are written with the same base.
Remember that logarithms are defined only for positive real numbers. Check your results in the original equation. You may have obtained a result that gives a logarithm of zero or a negative number.
Another strategy to use to solve logarithmic equations is to condense sums or differences into a single logarithm.
When there are logarithms on both sides, we condense each side into a single logarithm. Remember to use the Power Property as needed.

### Solve Exponential Equations Using Logarithms

In the section on exponential functions, we solved some equations by writing both sides of the equation with the same base. Next we wrote a new equation by setting the exponents equal.
It is not always possible or convenient to write the expressions with the same base. In that case we often take the common logarithm or natural logarithm of both sides once the exponential is isolated.
When we take the logarithm of both sides we will get the same result whether we use the common or the natural logarithm (try using the natural log in the last example. Did you get the same result?) When the exponential has base e, we use the natural logarithm.

### Use Exponential Models in Applications

In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.
We will again use the Compound Interest Formulas and so we list them here for reference.
We have seen that growth and decay are modeled by exponential functions. For growth and decay we use the formula  Exponential growth has a positive rate of growth or growth constant, , and exponential decay has a negative rate of growth or decay constant, k.
We can now solve applications that give us enough information to determine the rate of growth. We can then use that rate of growth to predict other situations.
Radioactive substances decay or decompose according to the exponential decay formula. The amount of time it takes for the substance to decay to half of its original amount is called the half-life of the substance.
Similar to the previous example, we can use the given information to determine the constant of decay, and then use that constant to answer other questions.

### Key Concepts

1. One-to-One Property of Logarithmic Equations: For  and  is any real number:

2. Compound Interest:
For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:

3. Exponential Growth and Decay: For an original amount,  that grows or decays at a rate, r, for a certain time t, the final amount, A, is 

### Section Exercises


### Practice Makes Perfect

Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
In the following exercises, solve each equation.
In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.
Use Exponential Models in Applications
In the following exercises, solve.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?

### Chapter Review Exercises


### Finding Composite and Inverse Functions

Find and Evaluate Composite Functions
In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).
In the following exercises, evaluate the composition.
Determine Whether a Function is One-to-One
In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
Find the Inverse of a Function
In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.
In the following exercise, graph the inverse of the one-to-one function shown.
In the following exercises, verify that the functions are inverse functions.
In the following exercises, find the inverse of each function.

### Evaluate and Graph Exponential Functions

Graph Exponential Functions
In the following exercises, graph each of the following functions.
Solve Exponential Equations
In the following exercises, solve each equation.
Use Exponential Models in Applications
In the following exercises, solve.

### Evaluate and Graph Logarithmic Functions

Convert Between Exponential and Logarithmic Form
In the following exercises, convert from exponential to logarithmic form.
In the following exercises, convert each logarithmic equation to exponential form.
Evaluate Logarithmic Functions
In the following exercises, solve for x.
In the following exercises, find the exact value of each logarithm without using a calculator.
Graph Logarithmic Functions
In the following exercises, graph each logarithmic function.
Solve Logarithmic Equations
In the following exercises, solve each logarithmic equation.
Use Logarithmic Models in Applications

### Use the Properties of Logarithms

Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.
In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
Use the Change-of-Base Formula
In the following exercises, rounding to three decimal places, approximate each logarithm.

### Solve Exponential and Logarithmic Equations

Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
Use Exponential Models in Applications

### Practice Test

In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.