Precalculus Examples

$4 \ln (\sqrt{x^{3}} - 5) \ln (\sqrt[5]{y^{4}})$

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{y^{4}}$ as $y^{\frac{4}{5}}$ .

$4 \ln (x^{\frac{3}{2}} - 5) \ln (y^{\frac{4}{5}})$

Step 3

Expand $\ln (y^{\frac{4}{5}})$ by moving $\frac{4}{5}$ outside the logarithm.

$4 \ln (x^{\frac{3}{2}} - 5) (\frac{4}{5} \ln (y))$

Step 4

Combine $\frac{4}{5}$ and $\ln (y)$ .

$4 \ln (x^{\frac{3}{2}} - 5) \frac{4 \ln (y)}{5}$

Step 5

Multiply $4 \ln (x^{\frac{3}{2}} - 5) \frac{4 \ln (y)}{5}$ .

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Step 5.1

Combine $\frac{4 \ln (y)}{5}$ and $4$ .

$\frac{4 \ln (y) \cdot 4}{5} \ln (x^{\frac{3}{2}} - 5)$

Step 5.2

Multiply $4$ by $4$ .

$\frac{16 \ln (y)}{5} \ln (x^{\frac{3}{2}} - 5)$

Step 5.3

Combine $\frac{16 \ln (y)}{5}$ and $\ln (x^{\frac{3}{2}} - 5)$ .

$\frac{16 \ln (y) \ln (x^{\frac{3}{2}} - 5)}{5}$