1. Objectives Solve exponential and logarithmic equations and equalities.
Solve problems involving exponential and logarithmic equations.

2. Vocabulary
exponential equation
logarithmic equation

3. An exponential equation is an equation containing one or more expressions that have a variable as an exponent. To solve exponential equations:
Try writing them so that the bases are all the same.
Take the logarithm of both sides.

4. When you use a rounded number in a check, the result will not be exact, but it should be reasonable.
Helpful Hint

5. Check It Out! Example 1a
Solve and check.
32x = 27

6. Check It Out! Example 1b
Solve and check.
7–x = 21

7. Check It Out! Example 1c
Solve and check.
23x = 15

8. Check It Out! Example 2
You receive one penny on the first day, and then triple that (3 cents) on the second day, and so on for a month. On what day would you receive a least a million dollars.

9. A logarithmic equation is an equation with a logarithmic expression that contains a variable. You can solve logarithmic equations by using the properties of logarithms.

10. Review the properties of logarithms from Lesson 7-4.
Remember!

11. Check It Out! Example 3a
Solve.
3 = log 8 + 3log x

12. Check It Out! Example 3b
Solve.
2log x – log 4 = 0

13. Watch out for calculated solutions that are not solutions of the original equation.
Caution

14. Check It Out! Example 4a
Use a table and graph to solve 2x = 4x – 1.
Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.
In the table, find the x-values where Y1 is equal to Y2.
In the graph, find the x-value at the point of intersection.
The solution is x = 2.

15. Check It Out! Example 4b
Use a table and graph to solve 2x > 4x – 1.
Use a graphing calculator. Enter 2x as Y1 and 4(x – 1) as Y2.
In the table, find the x-values where Y1 is greater than Y2.
In the graph, find the x-value at the point of intersection.
The solution is x < 2.

16. Check It Out! Example 4c
Use a table and graph to solve log x2 = 6.
Use a graphing calculator. Enter log(x2) as Y1 and 6 as Y2.
In the table, find the x-values where Y1 is equal to Y2.
In the graph, find the x-value at the point of intersection.
The solution is x = 1000.