Precalculus Examples

$\log (\frac{\sqrt{3 x}}{7 y^{3}})$

Step 1

Rewrite $\log (\frac{\sqrt{3 x}}{7 y^{3}})$ as $\log (\sqrt{3 x}) - \log (7 y^{3})$ .

$\log (\sqrt{3 x}) - \log (7 y^{3})$

Step 2

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

$\log ({(3 x)}^{\frac{1}{2}}) - \log (7 y^{3})$

Step 3

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

$\frac{1}{2} \log (3 x) - \log (7 y^{3})$

Step 4

Rewrite $\log (3 x)$ as $\log (3) + \log (x)$ .

$\frac{1}{2} (\log (3) + \log (x)) - \log (7 y^{3})$

Step 5

Rewrite $\log (7 y^{3})$ as $\log (7) + \log (y^{3})$ .

$\frac{1}{2} (\log (3) + \log (x)) - (\log (7) + \log (y^{3}))$

Step 6

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

$\frac{1}{2} (\log (3) + \log (x)) - (\log (7) + 3 \log (y))$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{2} \log (3) + \frac{1}{2} \log (x) - (\log (7) + 3 \log (y))$

Step 7.2

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

$\frac{\log (3)}{2} + \frac{1}{2} \log (x) - (\log (7) + 3 \log (y))$

Step 7.3

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

$\frac{\log (3)}{2} + \frac{\log (x)}{2} - (\log (7) + 3 \log (y))$

Step 7.4

Apply the distributive property.

$\frac{\log (3)}{2} + \frac{\log (x)}{2} - \log (7) - (3 \log (y))$

Step 7.5

Multiply $3$ by $- 1$ .

$\frac{\log (3)}{2} + \frac{\log (x)}{2} - \log (7) - 3 \log (y)$