Evaluate logarithms

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

Now consider solving ${\mathrm{log}}_{7}49$ and ${\mathrm{log}}_{3}27$ mentally.

We ask, “To what exponent must 7 be raised in order to get 49?” We know ${7}^{2}=49$ . Therefore, ${\mathrm{log}}_{7}49=2$
We ask, “To what exponent must 3 be raised in order to get 27?” We know ${3}^{3}=27$ . Therefore, ${\mathrm{log}}_{3}27=3$

Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let’s evaluate ${\mathrm{log}}_{\frac{2}{3}}\frac{4}{9}$ 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 ${\left(\frac{2}{3}\right)}^{2}=\frac{4}{9}$ . Therefore, ${\mathrm{log}}_{\frac{2}{3}}\left(\frac{4}{9}\right)=2$ .

How To: Given a logarithm of the form $y={\mathrm{log}}_{b}\left(x\right)$ , evaluate it mentally.

Rewrite the argument x as a power of b: ${b}^{y}=x$ .
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 $y={\mathrm{log}}_{4}\left(64\right)$ 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

${4}^{3}=64$

Therefore,

l o g_{4} (64) = 3

$\mathrm{log}{}_{4}\left(64\right)=3$

Try It 3

Solve $y={\mathrm{log}}_{121}\left(11\right)$ without using a calculator.

Solution

Example 4: Evaluating the Logarithm of a Reciprocal

Evaluate $y={\mathrm{log}}_{3}\left(\frac{1}{27}\right)$ 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

{\begin{matrix} 3^{- 3} = \frac{1}{3^{3}} = \frac{1}{27} \end{matrix}

$\begin{cases}{3}^{-3}=\frac{1}{{3}^{3}}\hfill \\ =\frac{1}{27}\hfill \end{cases}$

Therefore, ${\mathrm{log}}_{3}\left(\frac{1}{27}\right)=-3$ .

Try It 4

Evaluate $y={\mathrm{log}}_{2}\left(\frac{1}{32}\right)$ without using a calculator.

Solution

Logarithmic Functions

How To: Given a logarithm of the form $y={\mathrm{log}}_{b}\left(x\right)$ , evaluate it mentally.

Example 3: Solving Logarithms Mentally

Solution

Try It 3

Example 4: Evaluating the Logarithm of a Reciprocal

Solution

Try It 4

Candela Citations

Logarithmic Functions