Precalculus Examples

$\frac{\log_{2} (27)}{- 3}$

Step 1

Rewrite $\log_{2} (27)$ as $\log_{2} (3^{3})$ .

$\frac{\log_{2} (3^{3})}{- 3}$

Step 2

Expand $\log_{2} (3^{3})$ by moving $3$ outside the logarithm.

$\frac{3 \log_{2} (3)}{- 3}$

Step 3

Cancel the common factor of $3$ and $- 3$ .

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Step 3.1

Factor $3$ out of $3 \log_{2} (3)$ .

$\frac{3 (\log_{2} (3))}{- 3}$

Step 3.2

Move the negative one from the denominator of $\frac{\log_{2} (3)}{- 1}$ .

$- 1 \cdot \log_{2} (3)$

Step 4

Rewrite $- 1 \log_{2} (3)$ as $- \log_{2} (3)$ .

$- \log_{2} (3)$

Step 5

Rewrite $\log_{2} (3)$ using the change of base formula.

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Step 5.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$- \log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 5.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$- \frac{\log (3)}{\log (2)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{\log (3)}{\log (2)}$

Decimal Form:

$- 1.58496250 \dots$