Precalculus Examples

$\frac{2 \ln (\sqrt{8})}{3}$

Step 1

Simplify $2 \ln (\sqrt{8})$ by moving $2$ inside the logarithm.

$\frac{\ln ({\sqrt{8}}^{2})}{3}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.1.1

Factor $4$ out of $8$ .

$\frac{\ln ({\sqrt{4 (2)}}^{2})}{3}$

Step 2.1.2

Rewrite $4$ as $2^{2}$ .

$\frac{\ln ({\sqrt{2^{2} \cdot 2}}^{2})}{3}$

Step 2.2

Pull terms out from under the radical.

$\frac{\ln ({(2 \sqrt{2})}^{2})}{3}$

Step 2.3

Apply the product rule to $2 \sqrt{2}$ .

$\frac{\ln (2^{2} {\sqrt{2}}^{2})}{3}$

Step 2.4

Raise $2$ to the power of $2$ .

$\frac{\ln (4 {\sqrt{2}}^{2})}{3}$

Step 2.5

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 2.5.1

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

$\frac{\ln (4 {(2^{\frac{1}{2}})}^{2})}{3}$

Step 2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\ln (4 \cdot 2^{\frac{1}{2} \cdot 2})}{3}$

Step 2.5.3

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

$\frac{\ln (4 \cdot 2^{\frac{2}{2}})}{3}$

Step 2.5.4

Cancel the common factor of $2$ .

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Step 2.5.4.1

Cancel the common factor.

$\frac{\ln (4 \cdot 2^{\frac{2}{2}})}{3}$

Step 2.5.4.2

Rewrite the expression.

$\frac{\ln (4 \cdot 2^{1})}{3}$

Step 2.5.5

Evaluate the exponent.

$\frac{\ln (4 \cdot 2)}{3}$

Step 2.6

Multiply $4$ by $2$ .

$\frac{\ln (8)}{3}$

Step 3

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

$\frac{\ln (2^{3})}{3}$

Step 4

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

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

Step 5

Cancel the common factor of $3$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $\ln (2)$ by $1$ .

$\ln (2)$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\ln (2)$

Decimal Form:

$0.69314718 \dots$