Precalculus Examples

$\ln (\frac{\sqrt{x - 1}}{\sqrt[3]{2 x - 3}})$

Step 1

Rewrite $\ln (\frac{\sqrt{x - 1}}{\sqrt[3]{2 x - 3}})$ as $\ln (\sqrt{x - 1}) - \ln (\sqrt[3]{2 x - 3})$ .

$\ln (\sqrt{x - 1}) - \ln (\sqrt[3]{2 x - 3})$

Step 2

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

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

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x - 3}$ as ${(2 x - 3)}^{\frac{1}{3}}$ .

$\ln ({(x - 1)}^{\frac{1}{2}}) - \ln ({(2 x - 3)}^{\frac{1}{3}})$

Step 4

Expand $\ln ({(x - 1)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (x - 1) - \ln ({(2 x - 3)}^{\frac{1}{3}})$

Step 5

Expand $\ln ({(2 x - 3)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{2} \ln (x - 1) - (\frac{1}{3} \ln (2 x - 3))$

Step 6

Combine $\frac{1}{3}$ and $\ln (2 x - 3)$ .

$\frac{1}{2} \ln (x - 1) - \frac{\ln (2 x - 3)}{3}$

Step 7

Combine $\frac{1}{2}$ and $\ln (x - 1)$ .

$\frac{\ln (x - 1)}{2} - \frac{\ln (2 x - 3)}{3}$

Step 8

To write $\frac{\ln (x - 1)}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\ln (x - 1)}{2} \cdot \frac{3}{3} - \frac{\ln (2 x - 3)}{3}$

Step 9

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

$\frac{\ln (x - 1)}{2} \cdot \frac{3}{3} - \frac{\ln (2 x - 3)}{3} \cdot \frac{2}{2}$

Step 10

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{\ln (x - 1)}{2}$ by $\frac{3}{3}$ .

$\frac{\ln (x - 1) \cdot 3}{2 \cdot 3} - \frac{\ln (2 x - 3)}{3} \cdot \frac{2}{2}$

Step 10.2

Multiply $2$ by $3$ .

$\frac{\ln (x - 1) \cdot 3}{6} - \frac{\ln (2 x - 3)}{3} \cdot \frac{2}{2}$

Step 10.3

Multiply $\frac{\ln (2 x - 3)}{3}$ by $\frac{2}{2}$ .

$\frac{\ln (x - 1) \cdot 3}{6} - \frac{\ln (2 x - 3) \cdot 2}{3 \cdot 2}$

Step 10.4

Multiply $3$ by $2$ .

$\frac{\ln (x - 1) \cdot 3}{6} - \frac{\ln (2 x - 3) \cdot 2}{6}$

Step 11

Combine the numerators over the common denominator.

$\frac{\ln (x - 1) \cdot 3 - \ln (2 x - 3) \cdot 2}{6}$

Step 12

Simplify the numerator.

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Step 12.1

Move $3$ to the left of $\ln (x - 1)$ .

$\frac{3 \cdot \ln (x - 1) - \ln (2 x - 3) \cdot 2}{6}$

Step 12.2

Multiply $2$ by $- 1$ .

$\frac{3 \ln (x - 1) - 2 \ln (2 x - 3)}{6}$