Precalculus Examples

$\log (\frac{x^{3} \cdot \sqrt{y^{5}}}{10000 \cdot \sqrt{z^{3}}})$

Step 1

Rewrite $\log (\frac{x^{3} \cdot \sqrt{y^{5}}}{10000 \cdot \sqrt{z^{3}}})$ as $\log (x^{3} \cdot \sqrt{y^{5}}) - \log (10000 \cdot \sqrt{z^{3}})$ .

$\log (x^{3} \cdot \sqrt{y^{5}}) - \log (10000 \cdot \sqrt{z^{3}})$

Step 2

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

$\log (x^{3} \cdot y^{\frac{5}{2}}) - \log (10000 \cdot \sqrt{z^{3}})$

Step 3

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

$\log (x^{3} \cdot y^{\frac{5}{2}}) - \log (10000 \cdot z^{\frac{3}{2}})$

Step 4

Rewrite $\log (x^{3} \cdot y^{\frac{5}{2}})$ as $\log (x^{3}) + \log (y^{\frac{5}{2}})$ .

$\log (x^{3}) + \log (y^{\frac{5}{2}}) - \log (10000 \cdot z^{\frac{3}{2}})$

Step 5

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

$3 \log (x) + \log (y^{\frac{5}{2}}) - \log (10000 \cdot z^{\frac{3}{2}})$

Step 6

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

$3 \log (x) + \frac{5}{2} \log (y) - \log (10000 \cdot z^{\frac{3}{2}})$

Step 7

Rewrite $\log (10000 \cdot z^{\frac{3}{2}})$ as $\log (10000) + \log (z^{\frac{3}{2}})$ .

$3 \log (x) + \frac{5}{2} \log (y) - (\log (10000) + \log (z^{\frac{3}{2}}))$

Step 8

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

$3 \log (x) + \frac{5}{2} \log (y) - (\log (10000) + \frac{3}{2} \log (z))$

Step 9

Logarithm base $10$ of $10000$ is $4$ .

$3 \log (x) + \frac{5}{2} \log (y) - (4 + \frac{3}{2} \log (z))$

Step 10

Simplify each term.

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Step 10.1

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

$3 \log (x) + \frac{5 \log (y)}{2} - (4 + \frac{3}{2} \log (z))$

Step 10.2

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

$3 \log (x) + \frac{5 \log (y)}{2} - (4 + \frac{3 \log (z)}{2})$

Step 10.3

Apply the distributive property.

$3 \log (x) + \frac{5 \log (y)}{2} - 1 \cdot 4 - \frac{3 \log (z)}{2}$

Step 10.4

Multiply $- 1$ by $4$ .

$3 \log (x) + \frac{5 \log (y)}{2} - 4 - \frac{3 \log (z)}{2}$