Precalculus Examples

$\log (\frac{{(2 x - 3)}^{2} {(x - 1)}^{5}}{\sqrt[3]{3 x - 4}})$

Rewrite $\log (\frac{{(2 x - 3)}^{2} {(x - 1)}^{5}}{\sqrt[3]{3 x - 4}})$ as $\log ({(2 x - 3)}^{2} {(x - 1)}^{5}) - \log (\sqrt[3]{3 x - 4})$ .

$\log ({(2 x - 3)}^{2} {(x - 1)}^{5}) - \log (\sqrt[3]{3 x - 4})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{3 x - 4}$ as ${(3 x - 4)}^{\frac{1}{3}}$ .

$\log ({(2 x - 3)}^{2} {(x - 1)}^{5}) - \log ({(3 x - 4)}^{\frac{1}{3}})$

Expand $\log ({(3 x - 4)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log ({(2 x - 3)}^{2} {(x - 1)}^{5}) - (\frac{1}{3} \log (3 x - 4))$

Combine $\frac{1}{3}$ and $\log (3 x - 4)$ .

$\log ({(2 x - 3)}^{2} {(x - 1)}^{5}) - \frac{\log (3 x - 4)}{3}$

Rewrite $\log ({(2 x - 3)}^{2} {(x - 1)}^{5})$ as $\log ({(2 x - 3)}^{2}) + \log ({(x - 1)}^{5})$ .

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

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

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

Expand $\log ({(x - 1)}^{5})$ by moving $5$ outside the logarithm.

$2 \log (2 x - 3) + 5 \log (x - 1) - \frac{\log (3 x - 4)}{3}$

To write $2 \log (2 x - 3)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 \log (x - 1) + 2 \log (2 x - 3) \cdot \frac{3}{3} - \frac{\log (3 x - 4)}{3}$

Combine $2 \log (2 x - 3)$ and $\frac{3}{3}$ .

$5 \log (x - 1) + \frac{2 \log (2 x - 3) \cdot 3}{3} - \frac{\log (3 x - 4)}{3}$

Combine the numerators over the common denominator.

$5 \log (x - 1) + \frac{2 \log (2 x - 3) \cdot 3 - \log (3 x - 4)}{3}$

Multiply $3$ by $2$ .

$5 \log (x - 1) + \frac{6 \log (2 x - 3) - \log (3 x - 4)}{3}$

To write $5 \log (x - 1)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$5 \log (x - 1) \cdot \frac{3}{3} + \frac{6 \log (2 x - 3) - \log (3 x - 4)}{3}$

Combine $5 \log (x - 1)$ and $\frac{3}{3}$ .

$\frac{5 \log (x - 1) \cdot 3}{3} + \frac{6 \log (2 x - 3) - \log (3 x - 4)}{3}$

Combine the numerators over the common denominator.

$\frac{5 \log (x - 1) \cdot 3 + 6 \log (2 x - 3) - \log (3 x - 4)}{3}$

Multiply $3$ by $5$ .

$\frac{15 \log (x - 1) + 6 \log (2 x - 3) - \log (3 x - 4)}{3}$