Precalculus Examples

$\ln (\frac{x^{3}}{y^{6} \sqrt{z^{7}}})$

Step 1

Rewrite $\ln (\frac{x^{3}}{y^{6} \sqrt{z^{7}}})$ as $\ln (x^{3}) - \ln (y^{6} \sqrt{z^{7}})$ .

$\ln (x^{3}) - \ln (y^{6} \sqrt{z^{7}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{7}}$ as $z^{\frac{7}{2}}$ .

$\ln (x^{3}) - \ln (y^{6} z^{\frac{7}{2}})$

Step 3

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

$3 \ln (x) - \ln (y^{6} z^{\frac{7}{2}})$

Step 4

Rewrite $\ln (y^{6} z^{\frac{7}{2}})$ as $\ln (y^{6}) + \ln (z^{\frac{7}{2}})$ .

$3 \ln (x) - (\ln (y^{6}) + \ln (z^{\frac{7}{2}}))$

Step 5

Expand $\ln (y^{6})$ by moving $6$ outside the logarithm.

$3 \ln (x) - (6 \ln (y) + \ln (z^{\frac{7}{2}}))$

Step 6

Expand $\ln (z^{\frac{7}{2}})$ by moving $\frac{7}{2}$ outside the logarithm.

$3 \ln (x) - (6 \ln (y) + \frac{7}{2} \ln (z))$

Step 7

Simplify each term.

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Step 7.1

Combine $\frac{7}{2}$ and $\ln (z)$ .

$3 \ln (x) - (6 \ln (y) + \frac{7 \ln (z)}{2})$

Step 7.2

Apply the distributive property.

$3 \ln (x) - (6 \ln (y)) - \frac{7 \ln (z)}{2}$

Step 7.3

Multiply $6$ by $- 1$ .

$3 \ln (x) - 6 \ln (y) - \frac{7 \ln (z)}{2}$