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9-12 Integrated Math 3 Unit 4 Lesson 17

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Lesson 17

17.1 Warm-up 17.2 Activity 17.3 Activity Student Lesson Summary Glossary

Unit 4, Lesson 17

Logarithm Power Rule

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Integrated Math 3

17.1

Warm-up

For each equation, find the value of the missing terms by finding the value of the logarithms and comparing the values on each side of the equation.

The first one is done for you. Discuss with your partner why it is true.

17.2

Activity

Use the pattern you noticed about logarithms of expressions with an exponent to write a conjecture.
Assume the conjecture is true. Rewrite each expression using your conjecture, then find the value of the expression.
If and , find the values of each logarithm. Explain or show your reasoning.

17.3

Activity

Let's work through some steps of a proof for your conjecture.

Start with the equation:

Rewrite the equation as a logarithm, and circle your answer to use later.
Raise each side of the original equation to the power of .
Combine the exponents on the left side of the equation so that the left side is written with a single base.
Rewrite the last equation as a logarithm with a base of .
Use your circled equation to replace any in that equation with an equivalent logarithm.

Student Lesson Summary

The power rule for logarithms allows us to rewrite logarithms with values raised to powers. The power rule states that

For example, .

Thinking about logarithms in relation to exponents, this may make more sense. We learned in an earlier course that

By rewriting parts of that equation into their logarithm form, we can combine the pieces to prove the power rule.

A logarithm with an argument raised to a power is equivalent to the power multiplied by the logarithm of the argument with a power of 1.