Precalculus Examples

$\log_{b} (\frac{x^{\frac{1}{6}} y^{\frac{1}{3}}}{z^{7}})$

Step 1

Rewrite $\log_{b} (\frac{x^{\frac{1}{6}} y^{\frac{1}{3}}}{z^{7}})$ as $\log_{b} (x^{\frac{1}{6}} y^{\frac{1}{3}}) - \log_{b} (z^{7})$ .

$\log_{b} (x^{\frac{1}{6}} y^{\frac{1}{3}}) - \log_{b} (z^{7})$

Step 2

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

$\log_{b} (x^{\frac{1}{6}} y^{\frac{1}{3}}) - (7 \log_{b} (z))$

Step 3

Multiply $7$ by $- 1$ .

$\log_{b} (x^{\frac{1}{6}} y^{\frac{1}{3}}) - 7 \log_{b} (z)$

Step 4

Rewrite $\log_{b} (x^{\frac{1}{6}} y^{\frac{1}{3}})$ as $\log_{b} (x^{\frac{1}{6}}) + \log_{b} (y^{\frac{1}{3}})$ .

$\log_{b} (x^{\frac{1}{6}}) + \log_{b} (y^{\frac{1}{3}}) - 7 \log_{b} (z)$

Step 5

Expand $\log_{b} (x^{\frac{1}{6}})$ by moving $\frac{1}{6}$ outside the logarithm.

$\frac{1}{6} \log_{b} (x) + \log_{b} (y^{\frac{1}{3}}) - 7 \log_{b} (z)$

Step 6

Expand $\log_{b} (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{6} \log_{b} (x) + \frac{1}{3} \log_{b} (y) - 7 \log_{b} (z)$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{1}{6}$ and $\log_{b} (x)$ .

$\frac{\log_{b} (x)}{6} + \frac{1}{3} \log_{b} (y) - 7 \log_{b} (z)$

Step 7.2

Combine $\frac{1}{3}$ and $\log_{b} (y)$ .

$\frac{\log_{b} (x)}{6} + \frac{\log_{b} (y)}{3} - 7 \log_{b} (z)$