Precalculus Examples

$\log_{6} (\frac{4 x^{3}}{\sqrt{y} {(x - 2)}^{6}})$

Rewrite $\log_{6} (\frac{4 x^{3}}{\sqrt{y} {(x - 2)}^{6}})$ as $\log_{6} (4 x^{3}) - \log_{6} (\sqrt{y} {(x - 2)}^{6})$ .

$\log_{6} (4 x^{3}) - \log_{6} (\sqrt{y} {(x - 2)}^{6})$

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

$\log_{6} (4 x^{3}) - \log_{6} (y^{\frac{1}{2}} {(x - 2)}^{6})$

Rewrite $\log_{6} (4 x^{3})$ as $\log_{6} (4) + \log_{6} (x^{3})$ .

$\log_{6} (4) + \log_{6} (x^{3}) - \log_{6} (y^{\frac{1}{2}} {(x - 2)}^{6})$

Expand $\log_{6} (x^{3})$ by moving $3$ outside the logarithm.

$\log_{6} (4) + 3 \log_{6} (x) - \log_{6} (y^{\frac{1}{2}} {(x - 2)}^{6})$

Rewrite $\log_{6} (y^{\frac{1}{2}} {(x - 2)}^{6})$ as $\log_{6} (y^{\frac{1}{2}}) + \log_{6} ({(x - 2)}^{6})$ .

$\log_{6} (4) + 3 \log_{6} (x) - (\log_{6} (y^{\frac{1}{2}}) + \log_{6} ({(x - 2)}^{6}))$

Expand $\log_{6} (y^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{6} (4) + 3 \log_{6} (x) - (\frac{1}{2} \log_{6} (y) + \log_{6} ({(x - 2)}^{6}))$

Expand $\log_{6} ({(x - 2)}^{6})$ by moving $6$ outside the logarithm.

$\log_{6} (4) + 3 \log_{6} (x) - (\frac{1}{2} \log_{6} (y) + 6 \log_{6} (x - 2))$

Simplify each term.

Tap for more steps...

Rewrite $\log_{6} (4)$ as $\log_{6} (2^{2})$ .

$\log_{6} (2^{2}) + 3 \log_{6} (x) - (\frac{1}{2} \log_{6} (y) + 6 \log_{6} (x - 2))$

Expand $\log_{6} (2^{2})$ by moving $2$ outside the logarithm.

$2 \log_{6} (2) + 3 \log_{6} (x) - (\frac{1}{2} \log_{6} (y) + 6 \log_{6} (x - 2))$

Combine $\frac{1}{2}$ and $\log_{6} (y)$ .

$2 \log_{6} (2) + 3 \log_{6} (x) - (\frac{\log_{6} (y)}{2} + 6 \log_{6} (x - 2))$

To write $6 \log_{6} (x - 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \log_{6} (2) + 3 \log_{6} (x) - (\frac{\log_{6} (y)}{2} + 6 \log_{6} (x - 2) \cdot \frac{2}{2})$

Combine $6 \log_{6} (x - 2)$ and $\frac{2}{2}$ .

$2 \log_{6} (2) + 3 \log_{6} (x) - (\frac{\log_{6} (y)}{2} + \frac{6 \log_{6} (x - 2) \cdot 2}{2})$

Combine the numerators over the common denominator.

$2 \log_{6} (2) + 3 \log_{6} (x) - \frac{\log_{6} (y) + 6 \log_{6} (x - 2) \cdot 2}{2}$

Multiply $2$ by $6$ .

$2 \log_{6} (2) + 3 \log_{6} (x) - \frac{\log_{6} (y) + 12 \log_{6} (x - 2)}{2}$