Precalculus Examples

$\log (\frac{\sqrt{3}}{y^{2} z^{3}})$

Step 1

Rewrite $\log (\frac{\sqrt{3}}{y^{2} z^{3}})$ as $\log (\sqrt{3}) - \log (y^{2} z^{3})$ .

$\log (\sqrt{3}) - \log (y^{2} z^{3})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3

Expand $\log (3^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

Combine $\frac{1}{2}$ and $\log (3)$ .

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

Step 5

Rewrite $\log (y^{2} z^{3})$ as $\log (y^{2}) + \log (z^{3})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

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

Step 8.2

Multiply $2$ by $- 1$ .

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

Step 8.3

Multiply $3$ by $- 1$ .

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