Precalculus Examples

$\log_{m} (\frac{\sqrt[3]{x^{2} y^{5}}}{z^{7}})$

Step 1

Rewrite $\log_{m} (\frac{\sqrt[3]{x^{2} y^{5}}}{z^{7}})$ as $\log_{m} (\sqrt[3]{x^{2} y^{5}}) - \log_{m} (z^{7})$ .

$\log_{m} (\sqrt[3]{x^{2} y^{5}}) - \log_{m} (z^{7})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2} y^{5}}$ as ${(x^{2} y^{5})}^{\frac{1}{3}}$ .

$\log_{m} ({(x^{2} y^{5})}^{\frac{1}{3}}) - \log_{m} (z^{7})$

Step 3

Expand $\log_{m} ({(x^{2} y^{5})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{m} (x^{2} y^{5}) - \log_{m} (z^{7})$

Step 4

Expand $\log_{m} (z^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{3} \log_{m} (x^{2} y^{5}) - (7 \log_{m} (z))$

Step 5

Multiply $7$ by $- 1$ .

$\frac{1}{3} \log_{m} (x^{2} y^{5}) - 7 \log_{m} (z)$

Step 6

Rewrite $\log_{m} (x^{2} y^{5})$ as $\log_{m} (x^{2}) + \log_{m} (y^{5})$ .

$\frac{1}{3} (\log_{m} (x^{2}) + \log_{m} (y^{5})) - 7 \log_{m} (z)$

Step 7

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

$\frac{1}{3} (2 \log_{m} (x) + \log_{m} (y^{5})) - 7 \log_{m} (z)$

Step 8

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

$\frac{1}{3} (2 \log_{m} (x) + 5 \log_{m} (y)) - 7 \log_{m} (z)$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$\frac{1}{3} (2 \log_{m} (x)) + \frac{1}{3} (5 \log_{m} (y)) - 7 \log_{m} (z)$

Step 9.2

Multiply $\frac{1}{3} (2 \log_{m} (x))$ .

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Step 9.2.1

Combine $2$ and $\frac{1}{3}$ .

$\frac{2}{3} \log_{m} (x) + \frac{1}{3} (5 \log_{m} (y)) - 7 \log_{m} (z)$

Step 9.2.2

Combine $\frac{2}{3}$ and $\log_{m} (x)$ .

$\frac{2 \log_{m} (x)}{3} + \frac{1}{3} (5 \log_{m} (y)) - 7 \log_{m} (z)$

Step 9.3

Multiply $\frac{1}{3} (5 \log_{m} (y))$ .

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Step 9.3.1

Combine $5$ and $\frac{1}{3}$ .

$\frac{2 \log_{m} (x)}{3} + \frac{5}{3} \log_{m} (y) - 7 \log_{m} (z)$

Step 9.3.2

Combine $\frac{5}{3}$ and $\log_{m} (y)$ .

$\frac{2 \log_{m} (x)}{3} + \frac{5 \log_{m} (y)}{3} - 7 \log_{m} (z)$