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

7-4 Logarithmic Equations and Inequalities Textbook page 478

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.

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.

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

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

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

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

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.

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.

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.

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.

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

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

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.

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.

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.

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.