Precalculus Examples

$\ln (\sqrt{8 t})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 t}$ as ${(8 t)}^{\frac{1}{2}}$ .

$\ln ({(8 t)}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(8 t)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (8 t)$

Step 3

Rewrite $\ln (8 t)$ as $\ln (8) + \ln (t)$ .

$\frac{1}{2} (\ln (8) + \ln (t))$

Step 4

Simplify each term.

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Step 4.1

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

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

Step 4.2

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

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Combine $3$ and $\frac{1}{2}$ .

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

Step 6.2

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

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

Step 7

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

$\frac{3 \ln (2)}{2} + \frac{\ln (t)}{2}$