Convert Between Logarithmic and Exponential

Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider $log_28$. We ask, "To what exponent must $2$ be raised in order to get 8?" Because we already know $2^3=8$, it follows that $log_28=3$.

Now consider solving $log_749$ and $log_327$ mentally.

• We ask, "To what exponent must 7 be raised in order to get 49?" We know $7^2=49$. Therefore, $log_749=2$
• We ask, "To what exponent must 3 be raised in order to get 27?" We know $3^3=27$. Therefore, $log_327=3$

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let's evaluate $log_{\frac{2}{3}} \frac{4}{9}$ mentally.

• We ask, "To what exponent must $\frac{2}{3}$ be raised in order to get $\frac{4}{9}$? " We know $2^2=4$ and $3^2=9$, so $(\frac{2}{3})^2= \frac{4}{9}$. Therefore, $log_{\frac{2}{3}}(\frac{4}{9})=2$.

HOW TO

Given a logarithm of the form $y=log_b(x)$, evaluate it mentally.

1. Rewrite the argument $x$ as a power of $b: b^y=x$.
2. 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 $y=log_4(64)$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $4^y=64$. Next, we ask, "To what exponent must 4 be raised in order to get 64?"

We know

$4^3=64$

Therefore,

$log_4(64)=3$

TRY IT #3

Solve $y=log_{121}$ (11) without using a calculator.

EXAMPLE 4

Evaluating the Logarithm of a Reciprocal

Evaluate $y=log_3(\frac{1}{27})$ without using a calculator.

Solution

First we rewrite the logarithm in exponential form: $3^y= \frac{1}{27}$. Next, we ask, "To what exponent must 3 be raised in order to get $\frac{1}{27}$?"

We know $3^3=27$, but what must we do to get the reciprocal, $\frac{1}{27}$? Recall from working with exponents that $b^{−a}= \frac{1}{b^a}$. We use this information to write

$3^{−3}=\frac{1}{3^3}$

$=\frac{1}{27}$

Therefore, $log_3(\frac{1}{27})=−3$.

TRY IT #4

Evaluate $y=log_2(\frac{1}{32})$ without using a calculator.