Precalculus Examples

$\log (\frac{x^{3}}{\sqrt{z y^{9}}})$

Step 1

Rewrite $\log (\frac{x^{3}}{\sqrt{z y^{9}}})$ as $\log (x^{3}) - \log (\sqrt{z y^{9}})$ .

$\log (x^{3}) - \log (\sqrt{z y^{9}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z y^{9}}$ as ${(z y^{9})}^{\frac{1}{2}}$ .

$\log (x^{3}) - \log ({(z y^{9})}^{\frac{1}{2}})$

Step 3

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

$3 \log (x) - \log ({(z y^{9})}^{\frac{1}{2}})$

Step 4

Expand $\log ({(z y^{9})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$3 \log (x) - (\frac{1}{2} \log (z y^{9}))$

Step 5

Combine $\frac{1}{2}$ and $\log (z y^{9})$ .

$3 \log (x) - \frac{\log (z y^{9})}{2}$

Step 6

Rewrite $\log (z y^{9})$ as $\log (z) + \log (y^{9})$ .

$3 \log (x) - \frac{\log (z) + \log (y^{9})}{2}$

Step 7

Expand $\log (y^{9})$ by moving $9$ outside the logarithm.

$3 \log (x) - \frac{\log (z) + 9 \log (y)}{2}$

Step 8

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

$3 \log (x) \cdot \frac{2}{2} - \frac{\log (z) + 9 \log (y)}{2}$

Step 9

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

$\frac{3 \log (x) \cdot 2}{2} - \frac{\log (z) + 9 \log (y)}{2}$

Step 10

Combine the numerators over the common denominator.

$\frac{3 \log (x) \cdot 2 - (\log (z) + 9 \log (y))}{2}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $2$ by $3$ .

$\frac{6 \log (x) - (\log (z) + 9 \log (y))}{2}$

Step 11.2

Apply the distributive property.

$\frac{6 \log (x) - \log (z) - (9 \log (y))}{2}$

Step 11.3

Multiply $9$ by $- 1$ .

$\frac{6 \log (x) - \log (z) - 9 \log (y)}{2}$