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# 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.

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

${4}^{3}=64\\$

Therefore,

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

### 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

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

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