Precalculus Examples

$\log (\frac{x^{3} \sqrt{y}}{z^{2} \cdot 8^{- \frac{2}{3}}})$

Step 1

Rewrite $\log (\frac{x^{3} \sqrt{y}}{z^{2} \cdot 8^{- \frac{2}{3}}})$ as $\log (x^{3} \sqrt{y}) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$ .

$\log (x^{3} \sqrt{y}) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$

Step 2

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

$\log (x^{3} y^{\frac{1}{2}}) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$

Step 3

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

$\log (x^{3}) + \log (y^{\frac{1}{2}}) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$

Step 4

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

$3 \log (x) + \log (y^{\frac{1}{2}}) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$

Step 5

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

$3 \log (x) + \frac{1}{2} \log (y) - \log (z^{2} \cdot 8^{- \frac{2}{3}})$

Step 6

Rewrite $\log (z^{2} \cdot 8^{- \frac{2}{3}})$ as $\log (z^{2}) + \log (8^{- \frac{2}{3}})$ .

$3 \log (x) + \frac{1}{2} \log (y) - (\log (z^{2}) + \log (8^{- \frac{2}{3}}))$

Step 7

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

$3 \log (x) + \frac{1}{2} \log (y) - (2 \log (z) + \log (8^{- \frac{2}{3}}))$

Step 8

Expand $\log (8^{- \frac{2}{3}})$ by moving $- \frac{2}{3}$ outside the logarithm.

$3 \log (x) + \frac{1}{2} \log (y) - (2 \log (z) - \frac{2}{3} \log (8))$

Step 9

Rewrite $\log (8)$ as $\log (2^{3})$ .

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

Step 10

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

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

Step 11

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

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

Step 11.2

Factor $3$ out of $3 \log (2)$ .

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

Step 11.3

Cancel the common factor.

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

Step 11.4

Rewrite the expression.

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

Step 12

Simplify each term.

Tap for more steps...

Step 12.1

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

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Multiply $2$ by $- 1$ .

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

Step 12.4

Multiply $- 2$ by $- 1$ .

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