Precalculus Examples

$\log_{8} (90)$

Step 1

Rewrite $\log_{8} (90)$ as $\log_{8} (2 \cdot 3^{2} \cdot 5)$ .

$\log_{8} (2 \cdot 3^{2} \cdot 5)$

Step 2

Rewrite $\log_{8} (2 \cdot 3^{2} \cdot 5)$ as $\log_{8} (2 \cdot 3^{2}) + \log_{8} (5)$ .

$\log_{8} (2 \cdot 3^{2}) + \log_{8} (5)$

Step 3

Simplify each term.

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Step 3.1

Rewrite $\log_{8} (2 \cdot 3^{2})$ as $\log_{8} (2) + \log_{8} (3^{2})$ .

$\log_{8} (2) + \log_{8} (3^{2}) + \log_{8} (5)$

Step 3.2

Simplify each term.

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Step 3.2.1

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$\frac{1}{3} + \log_{8} (3^{2}) + \log_{8} (5)$

Step 3.2.2

Expand $\log_{8} (3^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{3} + 2 \log_{8} (3) + \log_{8} (5)$