Precalculus Examples

$\log_{b} (\frac{p^{3} q^{5}}{m^{5} b^{9}})$

Step 1

Rewrite $\log_{b} (\frac{p^{3} q^{5}}{m^{5} b^{9}})$ as $\log_{b} (p^{3} q^{5}) - \log_{b} (m^{5} b^{9})$ .

$\log_{b} (p^{3} q^{5}) - \log_{b} (m^{5} b^{9})$

Step 2

Rewrite $\log_{b} (p^{3} q^{5})$ as $\log_{b} (p^{3}) + \log_{b} (q^{5})$ .

$\log_{b} (p^{3}) + \log_{b} (q^{5}) - \log_{b} (m^{5} b^{9})$

Step 3

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

$3 \log_{b} (p) + \log_{b} (q^{5}) - \log_{b} (m^{5} b^{9})$

Step 4

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

$3 \log_{b} (p) + 5 \log_{b} (q) - \log_{b} (m^{5} b^{9})$

Step 5

Rewrite $\log_{b} (m^{5} b^{9})$ as $\log_{b} (m^{5}) + \log_{b} (b^{9})$ .

$3 \log_{b} (p) + 5 \log_{b} (q) - (\log_{b} (m^{5}) + \log_{b} (b^{9}))$

Step 6

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

$3 \log_{b} (p) + 5 \log_{b} (q) - (5 \log_{b} (m) + \log_{b} (b^{9}))$

Step 7

Expand $\log_{b} (b^{9})$ by moving $9$ outside the logarithm.

$3 \log_{b} (p) + 5 \log_{b} (q) - (5 \log_{b} (m) + 9 \log_{b} (b))$

Step 8

Logarithm base $b$ of $b$ is $1$ .

$3 \log_{b} (p) + 5 \log_{b} (q) - (5 \log_{b} (m) + 9 \cdot 1)$

Step 9

Multiply $9$ by $1$ .

$3 \log_{b} (p) + 5 \log_{b} (q) - (5 \log_{b} (m) + 9)$

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

$3 \log_{b} (p) + 5 \log_{b} (q) - (5 \log_{b} (m)) - 1 \cdot 9$

Step 10.2

Multiply $5$ by $- 1$ .

$3 \log_{b} (p) + 5 \log_{b} (q) - 5 \log_{b} (m) - 1 \cdot 9$

Step 10.3

Multiply $- 1$ by $9$ .

$3 \log_{b} (p) + 5 \log_{b} (q) - 5 \log_{b} (m) - 9$