Evaluate logarithms
Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider . We ask, "To what exponent must 2 be raised in order to get 8?" Because we already know , it follows that .
Now consider solving and mentally.
- We ask, "To what exponent must 7 be raised in order to get 49?" We know . Therefore,
- We ask, "To what exponent must 3 be raised in order to get 27?" We know . Therefore,
Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate mentally.
- We ask, "To what exponent must be raised in order to get ? " We know and , so . Therefore, .
How To: Given a logarithm of the form , evaluate it mentally.
- Rewrite the argument x as a power of b: .
- Use previous knowledge of powers of b identify y by asking, "To what exponent should b be raised in order to get x?"
Example 3: Solving Logarithms Mentally
Solve without using a calculator.
Solution
First we rewrite the logarithm in exponential form: . Next, we ask, "To what exponent must 4 be raised in order to get 64?"
We know
Therefore,
Try It 3
Solve without using a calculator.
SolutionExample 4: Evaluating the Logarithm of a Reciprocal
Evaluate without using a calculator.
Solution
First we rewrite the logarithm in exponential form: . Next, we ask, "To what exponent must 3 be raised in order to get "?
We know , but what must we do to get the reciprocal, ? Recall from working with exponents that . We use this information to write
Therefore, .
Try It 4
Evaluate without using a calculator.
Solution