Precalculus Examples

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

Step 1

Rewrite $\log (y^{3} \sqrt{z})$ as $\log (y^{3}) + \log (\sqrt{z})$ .

$\frac{\log (y^{3}) + \log (\sqrt{z})}{x^{5}}$

Step 2

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

$\frac{\log (y^{3}) + \log (z^{\frac{1}{2}})}{x^{5}}$

Step 3

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

$\frac{3 \log (y) + \log (z^{\frac{1}{2}})}{x^{5}}$

Step 4

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

$\frac{3 \log (y) + \frac{1}{2} \log (z)}{x^{5}}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{3 \log (y) + \frac{\log (z)}{2}}{x^{5}}$

Step 5.2

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

$\frac{3 \log (y) \cdot \frac{2}{2} + \frac{\log (z)}{2}}{x^{5}}$

Step 5.3

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

$\frac{\frac{3 \log (y) \cdot 2}{2} + \frac{\log (z)}{2}}{x^{5}}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{\frac{3 \log (y) \cdot 2 + \log (z)}{2}}{x^{5}}$

Step 5.5

Multiply $2$ by $3$ .

$\frac{\frac{6 \log (y) + \log (z)}{2}}{x^{5}}$

Step 6

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 \log (y) + \log (z)}{2} \cdot \frac{1}{x^{5}}$

Step 7

Multiply $\frac{6 \log (y) + \log (z)}{2}$ by $\frac{1}{x^{5}}$ .

$\frac{6 \log (y) + \log (z)}{2 x^{5}}$