Convert Between Logarithmic and Exponential
Logarithms are the inverses of exponential functions. You will explore the relationship between an exponential and a logarithmic function. You will also explore the basic characteristics of a logarithmic function, including domain, range, and long-run behavior.
Evaluating 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 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.
HOW TO
Given a logarithm of the form , evaluate it mentally.
- Rewrite the argument as a power of .
- Use previous knowledge of powers of identify by asking, "To what exponent should be raised in order to get ?"
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 (11) without using a calculator.
EXAMPLE 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