Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\log (y x^{\frac{5}{2}})$ as $\log (y) + \log (x^{\frac{5}{2}})$ .

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

Step 5

Expand $\log (x^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

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

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{5}{2}$ and $\log (x)$ .

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

Step 6.2

Apply the distributive property.

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