Expanding and Condensing Logarithms
Finally, we will wrap up the properties of logarithms by learning how to expand and condense logarithms and use the change of base formula.
Using the Change-of-Base Formula for Logarithms
Most calculators can evaluate only common and natural logs. In order to evaluate logarithms with a base other than 10 or , we use the change-of-base formula to rewrite the logarithm as the quotient of logarithms of any other base; when using a calculator, we would change them to common or natural logs.
To derive the change-of-base formula, we use the one-to-one property and power rule for logarithms.
Given any positive real numbers , and , where and , we show
Let . By exponentiating both sides with base , we arrive at an exponential form, namely . It follows that
For example, to evaluate using a calculator, we must first rewrite the expression as a quotient of common or natural logs. We will use the common log.
THE CHANGE-OF-BASE FORMULA
The change-of-base formula can be used to evaluate a logarithm with any base.
For any positive real numbers , and , where and ,
It follows that the change-of-base formula can be used to rewrite a logarithm with any base as the quotient of common or natural logs.
and
HOW TO
Given a logarithm with the form , use the change-of-base formula to rewrite it as a quotient of logs with any positive base , where .
- Determine the new base , remembering that the common , has base 10 , and the natural , has base .
- Rewrite the log as a quotient using the change-of-base formula
EXAMPLE 13
Changing Logarithmic Expressions to Expressions Involving Only Natural Logs
Change to a quotient of natural logarithms.
Solution
Because we will be expressing as a quotient of natural logarithms, the new base, .
We rewrite the log as a quotient using the change-of-base formula. The numerator of the quotient will be the natural log with argument 3. The denominator of the quotient will be the natural log with argument 5.
TRY IT #13
Change to a quotient of natural logarithms.
Q&A
Can we change common logarithms to natural logarithms?
Yes. Remember that means . So, .
EXAMPLE 14
Using the Change-of-Base Formula with a Calculator
Evaluate using the change-of-base formula with a calculator.
Solution
According to the change-of-base formula, we can rewrite the log base 2 as a logarithm of any other base. Since our calculators can evaluate the natural log, we might choose to use the natural logarithm, which is the log base .
TRY IT #14
Evaluate using the change-of-base formula.