Precalculus Examples

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

Step 1

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

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

Step 2

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

$\log_{3} (x^{2} y^{3}) - \log_{3} (4 z^{\frac{1}{2}})$

Step 3

Rewrite $\log_{3} (x^{2} y^{3})$ as $\log_{3} (x^{2}) + \log_{3} (y^{3})$ .

$\log_{3} (x^{2}) + \log_{3} (y^{3}) - \log_{3} (4 z^{\frac{1}{2}})$

Step 4

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

$2 \log_{3} (x) + \log_{3} (y^{3}) - \log_{3} (4 z^{\frac{1}{2}})$

Step 5

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

$2 \log_{3} (x) + 3 \log_{3} (y) - \log_{3} (4 z^{\frac{1}{2}})$

Step 6

Rewrite $\log_{3} (4 z^{\frac{1}{2}})$ as $\log_{3} (4) + \log_{3} (z^{\frac{1}{2}})$ .

$2 \log_{3} (x) + 3 \log_{3} (y) - (\log_{3} (4) + \log_{3} (z^{\frac{1}{2}}))$

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

$2 \log_{3} (x) + 3 \log_{3} (y) - (\log_{3} (2^{2}) + \frac{1}{2} \log_{3} (z))$

Step 8.1.2

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

$2 \log_{3} (x) + 3 \log_{3} (y) - (2 \log_{3} (2) + \frac{1}{2} \log_{3} (z))$

Step 8.1.3

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

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

Step 8.2

Apply the distributive property.

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

Step 8.3

Multiply $2$ by $- 1$ .

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