Precalculus Examples

$5 \log (\frac{x^{2} \sqrt{z}}{9 y^{2}})$

Step 1

Rewrite $\log (\frac{x^{2} \sqrt{z}}{9 y^{2}})$ as $\log (x^{2} \sqrt{z}) - \log (9 y^{2})$ .

$5 (\log (x^{2} \sqrt{z}) - \log (9 y^{2}))$

Step 2

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

$5 (\log (x^{2} z^{\frac{1}{2}}) - \log (9 y^{2}))$

Step 3

Rewrite $\log (x^{2} z^{\frac{1}{2}})$ as $\log (x^{2}) + \log (z^{\frac{1}{2}})$ .

$5 (\log (x^{2}) + \log (z^{\frac{1}{2}}) - \log (9 y^{2}))$

Step 4

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

$5 (2 \log (x) + \log (z^{\frac{1}{2}}) - \log (9 y^{2}))$

Step 5

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

$5 (2 \log (x) + \frac{1}{2} \log (z) - \log (9 y^{2}))$

Step 6

Rewrite $\log (9 y^{2})$ as $\log (9) + \log (y^{2})$ .

$5 (2 \log (x) + \frac{1}{2} \log (z) - (\log (9) + \log (y^{2})))$

Step 7

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

$5 (2 \log (x) + \frac{1}{2} \log (z) - (\log (9) + 2 \log (y)))$

Step 8

Simplify each term.

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Step 8.1

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

$5 (2 \log (x) + \frac{\log (z)}{2} - (\log (9) + 2 \log (y)))$

Step 8.2

Simplify each term.

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Step 8.2.1

Rewrite $\log (9)$ as $\log (3^{2})$ .

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

Step 8.2.2

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

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

Step 8.3

Apply the distributive property.

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

Step 8.4

Multiply $2$ by $- 1$ .

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

Step 8.5

Multiply $2$ by $- 1$ .

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

Multiply $2$ by $5$ .

$10 \log (x) + 5 \frac{\log (z)}{2} + 5 (- 2 \log (3)) + 5 (- 2 \log (y))$

Step 10.2

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

$10 \log (x) + \frac{5 \log (z)}{2} + 5 (- 2 \log (3)) + 5 (- 2 \log (y))$

Step 10.3

Multiply $- 2$ by $5$ .

$10 \log (x) + \frac{5 \log (z)}{2} - 10 \log (3) + 5 (- 2 \log (y))$

Step 10.4

Multiply $- 2$ by $5$ .

$10 \log (x) + \frac{5 \log (z)}{2} - 10 \log (3) - 10 \log (y)$