Precalculus Examples

$\log_{2} (\frac{3 x (\sqrt[3]{y z})}{p^{3} q^{7}})$

Step 1

Rewrite $\log_{2} (\frac{3 x (\sqrt[3]{y z})}{p^{3} q^{7}})$ as $\log_{2} (3 x (\sqrt[3]{y z})) - \log_{2} (p^{3} q^{7})$ .

$\log_{2} (3 x (\sqrt[3]{y z})) - \log_{2} (p^{3} q^{7})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y z}$ as ${(y z)}^{\frac{1}{3}}$ .

$\log_{2} (3 x {(y z)}^{\frac{1}{3}}) - \log_{2} (p^{3} q^{7})$

Step 3

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

$\log_{2} (3 x) + \log_{2} ({(y z)}^{\frac{1}{3}}) - \log_{2} (p^{3} q^{7})$

Step 4

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

$\log_{2} (3) + \log_{2} (x) + \log_{2} ({(y z)}^{\frac{1}{3}}) - \log_{2} (p^{3} q^{7})$

Step 5

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

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} \log_{2} (y z) - \log_{2} (p^{3} q^{7})$

Step 6

Rewrite $\log_{2} (y z)$ as $\log_{2} (y) + \log_{2} (z)$ .

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} (\log_{2} (y) + \log_{2} (z)) - \log_{2} (p^{3} q^{7})$

Step 7

Rewrite $\log_{2} (p^{3} q^{7})$ as $\log_{2} (p^{3}) + \log_{2} (q^{7})$ .

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} (\log_{2} (y) + \log_{2} (z)) - (\log_{2} (p^{3}) + \log_{2} (q^{7}))$

Step 8

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

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} (\log_{2} (y) + \log_{2} (z)) - (3 \log_{2} (p) + \log_{2} (q^{7}))$

Step 9

Expand $\log_{2} (q^{7})$ by moving $7$ outside the logarithm.

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} (\log_{2} (y) + \log_{2} (z)) - (3 \log_{2} (p) + 7 \log_{2} (q))$

Step 10

Simplify each term.

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Step 10.1

Apply the distributive property.

$\log_{2} (3) + \log_{2} (x) + \frac{1}{3} \log_{2} (y) + \frac{1}{3} \log_{2} (z) - (3 \log_{2} (p) + 7 \log_{2} (q))$

Step 10.2

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

$\log_{2} (3) + \log_{2} (x) + \frac{\log_{2} (y)}{3} + \frac{1}{3} \log_{2} (z) - (3 \log_{2} (p) + 7 \log_{2} (q))$

Step 10.3

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

$\log_{2} (3) + \log_{2} (x) + \frac{\log_{2} (y)}{3} + \frac{\log_{2} (z)}{3} - (3 \log_{2} (p) + 7 \log_{2} (q))$

Step 10.4

Apply the distributive property.

$\log_{2} (3) + \log_{2} (x) + \frac{\log_{2} (y)}{3} + \frac{\log_{2} (z)}{3} - (3 \log_{2} (p)) - (7 \log_{2} (q))$

Step 10.5

Multiply $3$ by $- 1$ .

$\log_{2} (3) + \log_{2} (x) + \frac{\log_{2} (y)}{3} + \frac{\log_{2} (z)}{3} - 3 \log_{2} (p) - (7 \log_{2} (q))$

Step 10.6

Multiply $7$ by $- 1$ .

$\log_{2} (3) + \log_{2} (x) + \frac{\log_{2} (y)}{3} + \frac{\log_{2} (z)}{3} - 3 \log_{2} (p) - 7 \log_{2} (q)$