Precalculus Examples

$\ln (\sqrt{3 r^{2} s})$

Step 1

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

$\ln ({(3 r^{2} s)}^{\frac{1}{2}})$

Step 2

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

$\frac{1}{2} \ln (3 r^{2} s)$

Step 3

Rewrite $\ln (3 r^{2} s)$ as $\ln (3 r^{2}) + \ln (s)$ .

$\frac{1}{2} (\ln (3 r^{2}) + \ln (s))$

Step 4

Rewrite $\ln (3 r^{2})$ as $\ln (3) + \ln (r^{2})$ .

$\frac{1}{2} (\ln (3) + \ln (r^{2}) + \ln (s))$

Step 5

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

$\frac{1}{2} (\ln (3) + 2 \ln (r) + \ln (s))$

Step 6

Apply the distributive property.

$\frac{1}{2} \ln (3) + \frac{1}{2} (2 \ln (r)) + \frac{1}{2} \ln (s)$

Step 7

Simplify.

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Step 7.1

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

$\frac{\ln (3)}{2} + \frac{1}{2} (2 \ln (r)) + \frac{1}{2} \ln (s)$

Step 7.2

Cancel the common factor of $2$ .

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Step 7.2.1

Factor $2$ out of $2 \ln (r)$ .

$\frac{\ln (3)}{2} + \frac{1}{2} (2 (\ln (r))) + \frac{1}{2} \ln (s)$

Step 7.2.2

Cancel the common factor.

$\frac{\ln (3)}{2} + \frac{1}{2} (2 \ln (r)) + \frac{1}{2} \ln (s)$

Step 7.2.3

Rewrite the expression.

$\frac{\ln (3)}{2} + \ln (r) + \frac{1}{2} \ln (s)$

Step 7.3

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

$\frac{\ln (3)}{2} + \ln (r) + \frac{\ln (s)}{2}$