Precalculus Examples

$\log_{3} (90 \cdot 283)$

Step 1

Rewrite $\log_{3} (90 \cdot 283)$ as $\log_{3} (90) + \log_{3} (283)$ .

$\log_{3} (90) + \log_{3} (283)$

Step 2

Simplify each term.

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Step 2.1

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

$\log_{3} (2 \cdot 3^{2} \cdot 5) + \log_{3} (283)$

Step 2.2

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

$\log_{3} (2 \cdot 3^{2}) + \log_{3} (5) + \log_{3} (283)$

Step 2.3

Simplify each term.

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Step 2.3.1

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

$\log_{3} (2) + \log_{3} (3^{2}) + \log_{3} (5) + \log_{3} (283)$

Step 2.3.2

Use logarithm rules to move $2$ out of the exponent.

$\log_{3} (2) + 2 \log_{3} (3) + \log_{3} (5) + \log_{3} (283)$

Step 2.3.3

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (2) + 2 \cdot 1 + \log_{3} (5) + \log_{3} (283)$

Step 2.3.4

Multiply $2$ by $1$ .

$\log_{3} (2) + 2 + \log_{3} (5) + \log_{3} (283)$