Use your understanding of logarithms to find the value of each expression. Be prepared to explain your reasoning.
18.2
Activity
Changing the Logarithm Base
The change of base rule for logarithms states that, for any positive number (except 1),
Use the change of base rule to rewrite each of these logarithms so that .
Use your calculator to approximate the values of the logarithms in the previous question. Then check your approximation using exponents.
If the rule should work for any base, then it should also work to change the logarithms into base . Select one of the logarithms to change into a logarithm with base , then use your calculator to check that it has the same approximation.
18.3
Activity
Proving the Change of Base Rule
Here are some steps for a proof of the change of base rule: .
Start with the expression . Rewrite this expression in two ways.
Use the power rule for logarithms.
Rewrite the expression so that it does not use a power.
Set the two expressions you wrote equal to each other. How can you turn this equation into the change of base rule?
Student Lesson Summary
The change of baserule allows us to rewrite any logarithm to have any other positive value (except 1) for the base. We can use this equation to change the base: .
Many calculators can calculate logarithms only in base 10 () or base (). The change of base rule can be used to change any logarithm into one of these bases so we can find approximate values of any logarithm.
For example, we know that is between 2 and 3 because and , but it requires a lot of guessing and checking to get closer to an actual value when our calculator can't compute logarithms in base 2. Using the change of base rule allows us to rewrite the logarithm as , and this can be entered into most scientific calculators to get about 2.3219. We could have also changed into logarithms with base to get , which has the same approximate value.
Glossary
change of base rule (for logarithms)
A logarithmic expression can be rewritten using logarithms with any positive base other than 1 using the equation .
