Precalculus Examples

$\log_{b} (\frac{972}{5^{6}})$

Step 1

Rewrite $\log_{b} (\frac{972}{5^{6}})$ as $\log_{b} (972) - \log_{b} (5^{6})$ .

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

Step 2

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

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

Step 3

Multiply $6$ by $- 1$ .

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

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 4.3

Simplify each term.

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Step 4.3.1

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

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

Step 4.3.2

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

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