Precalculus Examples

$\ln (\frac{\sqrt[3]{x y}}{t^{\frac{5}{3}}})$

Step 1

Rewrite $\ln (\frac{\sqrt[3]{x y}}{t^{\frac{5}{3}}})$ as $\ln (\sqrt[3]{x y}) - \ln (t^{\frac{5}{3}})$ .

$\ln (\sqrt[3]{x y}) - \ln (t^{\frac{5}{3}})$

Step 2

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

$\ln ({(x y)}^{\frac{1}{3}}) - \ln (t^{\frac{5}{3}})$

Step 3

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

$\frac{1}{3} \ln (x y) - \ln (t^{\frac{5}{3}})$

Step 4

Expand $\ln (t^{\frac{5}{3}})$ by moving $\frac{5}{3}$ outside the logarithm.

$\frac{1}{3} \ln (x y) - (\frac{5}{3} \ln (t))$

Step 5

Combine $\frac{5}{3}$ and $\ln (t)$ .

$\frac{1}{3} \ln (x y) - \frac{5 \ln (t)}{3}$

Step 6

Rewrite $\ln (x y)$ as $\ln (x) + \ln (y)$ .

$\frac{1}{3} (\ln (x) + \ln (y)) - \frac{5 \ln (t)}{3}$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{1}{3} \ln (x) + \frac{1}{3} \ln (y) - \frac{5 \ln (t)}{3}$

Step 7.2

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

$\frac{\ln (x)}{3} + \frac{1}{3} \ln (y) - \frac{5 \ln (t)}{3}$

Step 7.3

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

$\frac{\ln (x)}{3} + \frac{\ln (y)}{3} - \frac{5 \ln (t)}{3}$