Precalculus Examples

$\log_{b} (\frac{3^{5} \cdot 2^{2}}{5^{6}})$

Step 1

Rewrite $\log_{b} (\frac{3^{5} \cdot 2^{2}}{5^{6}})$ as $\log_{b} (3^{5} \cdot 2^{2}) - \log_{b} (5^{6})$ .

$\log_{b} (3^{5} \cdot 2^{2}) - \log_{b} (5^{6})$

Step 2

Expand $\log_{b} (5^{6})$ by moving $6$ outside the logarithm.

$\log_{b} (3^{5} \cdot 2^{2}) - (6 \log_{b} (5))$

Step 3

Raise $3$ to the power of $5$ .

$\log_{b} (243 \cdot 2^{2}) - (6 \log_{b} (5))$

Step 4

Raise $2$ to the power of $2$ .

$\log_{b} (243 \cdot 4) - (6 \log_{b} (5))$

Step 5

Multiply $243$ by $4$ .

$\log_{b} (972) - (6 \log_{b} (5))$

Step 6

Multiply $6$ by $- 1$ .

$\log_{b} (972) - 6 \log_{b} (5)$

Step 7

Simplify each term.

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Step 7.1

Rewrite $\log_{b} (972)$ as $\log_{b} (2^{2} \cdot 3^{5})$ .

$\log_{b} (2^{2} \cdot 3^{5}) - 6 \log_{b} (5)$

Step 7.2

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

$\log_{b} (2^{2}) + \log_{b} (3^{5}) - 6 \log_{b} (5)$

Step 7.3

Simplify each term.

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Step 7.3.1

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

$2 \log_{b} (2) + \log_{b} (3^{5}) - 6 \log_{b} (5)$

Step 7.3.2

Expand $\log_{b} (3^{5})$ by moving $5$ outside the logarithm.

$2 \log_{b} (2) + 5 \log_{b} (3) - 6 \log_{b} (5)$