Precalculus Examples

$\log_{6} (\frac{x^{3}}{\sqrt{y z^{5}}})$

Step 1

Rewrite $\log_{6} (\frac{x^{3}}{\sqrt{y z^{5}}})$ as $\log_{6} (x^{3}) - \log_{6} (\sqrt{y z^{5}})$ .

$\log_{6} (x^{3}) - \log_{6} (\sqrt{y z^{5}})$

Step 2

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

$\log_{6} (x^{3}) - \log_{6} ({(y z^{5})}^{\frac{1}{2}})$

Step 3

Expand $\log_{6} (x^{3})$ by moving $3$ outside the logarithm.

$3 \log_{6} (x) - \log_{6} ({(y z^{5})}^{\frac{1}{2}})$

Step 4

Expand $\log_{6} ({(y z^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \log_{6} (x) - (\frac{1}{2} \log_{6} (y z^{5}))$

Step 5

Combine $\frac{1}{2}$ and $\log_{6} (y z^{5})$ .

$3 \log_{6} (x) - \frac{\log_{6} (y z^{5})}{2}$

Step 6

Rewrite $\log_{6} (y z^{5})$ as $\log_{6} (y) + \log_{6} (z^{5})$ .

$3 \log_{6} (x) - \frac{\log_{6} (y) + \log_{6} (z^{5})}{2}$

Step 7

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

$3 \log_{6} (x) - \frac{\log_{6} (y) + 5 \log_{6} (z)}{2}$

Step 8

To write $3 \log_{6} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 \log_{6} (x) \cdot \frac{2}{2} - \frac{\log_{6} (y) + 5 \log_{6} (z)}{2}$

Step 9

Combine $3 \log_{6} (x)$ and $\frac{2}{2}$ .

$\frac{3 \log_{6} (x) \cdot 2}{2} - \frac{\log_{6} (y) + 5 \log_{6} (z)}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{3 \log_{6} (x) \cdot 2 - (\log_{6} (y) + 5 \log_{6} (z))}{2}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $2$ by $3$ .

$\frac{6 \log_{6} (x) - (\log_{6} (y) + 5 \log_{6} (z))}{2}$

Step 11.2

Apply the distributive property.

$\frac{6 \log_{6} (x) - \log_{6} (y) - (5 \log_{6} (z))}{2}$

Step 11.3

Multiply $5$ by $- 1$ .

$\frac{6 \log_{6} (x) - \log_{6} (y) - 5 \log_{6} (z)}{2}$