Precalculus Examples

$\ln (\frac{m^{5} \sqrt[4]{b^{3}}}{p^{3} r})$

Step 1

Rewrite $\ln (\frac{m^{5} \sqrt[4]{b^{3}}}{p^{3} r})$ as $\ln (m^{5} \sqrt[4]{b^{3}}) - \ln (p^{3} r)$ .

$\ln (m^{5} \sqrt[4]{b^{3}}) - \ln (p^{3} r)$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{b^{3}}$ as $b^{\frac{3}{4}}$ .

$\ln (m^{5} b^{\frac{3}{4}}) - \ln (p^{3} r)$

Step 3

Rewrite $\ln (m^{5} b^{\frac{3}{4}})$ as $\ln (m^{5}) + \ln (b^{\frac{3}{4}})$ .

$\ln (m^{5}) + \ln (b^{\frac{3}{4}}) - \ln (p^{3} r)$

Step 4

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

$5 \ln (m) + \ln (b^{\frac{3}{4}}) - \ln (p^{3} r)$

Step 5

Expand $\ln (b^{\frac{3}{4}})$ by moving $\frac{3}{4}$ outside the logarithm.

$5 \ln (m) + \frac{3}{4} \ln (b) - \ln (p^{3} r)$

Step 6

Rewrite $\ln (p^{3} r)$ as $\ln (p^{3}) + \ln (r)$ .

$5 \ln (m) + \frac{3}{4} \ln (b) - (\ln (p^{3}) + \ln (r))$

Step 7

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

$5 \ln (m) + \frac{3}{4} \ln (b) - (3 \ln (p) + \ln (r))$

Step 8

Simplify each term.

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Step 8.1

Combine $\frac{3}{4}$ and $\ln (b)$ .

$5 \ln (m) + \frac{3 \ln (b)}{4} - (3 \ln (p) + \ln (r))$

Step 8.2

Apply the distributive property.

$5 \ln (m) + \frac{3 \ln (b)}{4} - (3 \ln (p)) - \ln (r)$

Step 8.3

Multiply $3$ by $- 1$ .

$5 \ln (m) + \frac{3 \ln (b)}{4} - 3 \ln (p) - \ln (r)$