Precalculus Examples

$\log (\frac{y^{4}}{x^{3} \sqrt{z^{2}}})$

Step 1

Rewrite $\log (\frac{y^{4}}{x^{3} \sqrt{z^{2}}})$ as $\log (y^{4}) - \log (x^{3} \sqrt{z^{2}})$ .

$\log (y^{4}) - \log (x^{3} \sqrt{z^{2}})$

Step 2

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

$\log (y^{4}) - \log (x^{3} z^{\frac{2}{2}})$

Step 3

Divide $2$ by $2$ .

$\log (y^{4}) - \log (x^{3} z^{1})$

Step 4

Expand $\log (y^{4})$ by moving $4$ outside the logarithm.

$4 \log (y) - \log (x^{3} z^{1})$

Step 5

Rewrite $\log (x^{3} z^{1})$ as $\log (x^{3}) + \log (z^{1})$ .

$4 \log (y) - (\log (x^{3}) + \log (z^{1}))$

Step 6

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

$4 \log (y) - (3 \log (x) + \log (z^{1}))$

Step 7

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

$4 \log (y) - (3 \log (x) + 1 \log (z))$

Step 8

Multiply $\log (z)$ by $1$ .

$4 \log (y) - (3 \log (x) + \log (z))$

Step 9

Simplify each term.

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Apply the distributive property.

$4 \log (y) - (3 \log (x)) - \log (z)$

Multiply $3$ by $- 1$ .

$4 \log (y) - 3 \log (x) - \log (z)$