1. Warm Up
Solve.
1. log16x =
2. logx8 = 3
3. log10,000 = x 3 2 64
1.1 4

2. 7-4 Logarithmic Equations and Inequalities
Textbook page 478

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. 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.

5. Solving by Rewriting as an Exponential
Solve log4(x+3) = 2
42 = x+3
16 = x+3
13 = x

6. Let’s rewrite the problem in exponential form.
Solution:
Let’s rewrite the problem in exponential form.
We’re finished !

7. Rewrite the problem in exponential form.
Solution:
Rewrite the problem in exponential form.

8. Solution: Example 3 Try setting this up like this:
Now rewrite in exponential form.

9. Solution: Example 4 First, we write the problem with a variable.
Now take it out of the logarithmic form
and write it in exponential form.

10. Solution: Example 5 First, we write the problem with a variable.
Now take it out of the exponential form
and write it in logarithmic form.

12. Basically, with logarithmic functions,
Finally, we want to take a look at the Property of Equality for Logarithmic Functions.
Basically, with logarithmic functions,
if the bases match on both sides of the equal sign , then simply set the arguments equal.

13. Since the bases are both ‘3’ we simply set the arguments equal.
Example 1
Solution:
Since the bases are both ‘3’ we simply set the arguments equal.

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

15. Solution: Example 2 Factor
Since the bases are both ‘8’ we simply set the arguments equal.
Factor
continued on the next page

16. Solution: Example 2 continued
It appears that we have 2 solutions here.
If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution.
Let’s end this lesson by taking a closer look at this.

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

19. 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.

20. 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.