Precalculus Examples

$\log_{2} (\frac{16 y^{3}}{x^{5} \sqrt{z}})$

Step 1

Rewrite $\log_{2} (\frac{16 y^{3}}{x^{5} \sqrt{z}})$ as $\log_{2} (16 y^{3}) - \log_{2} (x^{5} \sqrt{z})$ .

$\log_{2} (16 y^{3}) - \log_{2} (x^{5} \sqrt{z})$

Step 2

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

$\log_{2} (16 y^{3}) - \log_{2} (x^{5} z^{\frac{1}{2}})$

Step 3

Rewrite $\log_{2} (16 y^{3})$ as $\log_{2} (16) + \log_{2} (y^{3})$ .

$\log_{2} (16) + \log_{2} (y^{3}) - \log_{2} (x^{5} z^{\frac{1}{2}})$

Step 4

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

$\log_{2} (16) + 3 \log_{2} (y) - \log_{2} (x^{5} z^{\frac{1}{2}})$

Step 5

Logarithm base $2$ of $16$ is $4$ .

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

Step 6

Rewrite $\log_{2} (x^{5} z^{\frac{1}{2}})$ as $\log_{2} (x^{5}) + \log_{2} (z^{\frac{1}{2}})$ .

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

Step 7

Expand $\log_{2} (x^{5})$ by moving $5$ outside the logarithm.

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

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

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

Step 9.2

Apply the distributive property.

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

Step 9.3

Multiply $5$ by $- 1$ .

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