Precalculus Examples

$\ln (\sqrt{5 e^{3} x^{4}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 e^{3} x^{4}}$ as ${(5 e^{3} x^{4})}^{\frac{1}{2}}$ .

$\ln ({(5 e^{3} x^{4})}^{\frac{1}{2}})$

Step 2

Expand $\ln ({(5 e^{3} x^{4})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (5 e^{3} x^{4})$

Step 3

Rewrite $\ln (5 e^{3} x^{4})$ as $\ln (5 e^{3}) + \ln (x^{4})$ .

$\frac{1}{2} (\ln (5 e^{3}) + \ln (x^{4}))$

Step 4

Rewrite $\ln (5 e^{3})$ as $\ln (5) + \ln (e^{3})$ .

$\frac{1}{2} (\ln (5) + \ln (e^{3}) + \ln (x^{4}))$

Step 5

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

$\frac{1}{2} (\ln (5) + 3 \ln (e) + \ln (x^{4}))$

Step 6

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

$\frac{1}{2} (\ln (5) + 3 \ln (e) + 4 \ln (x))$

Step 7

The natural logarithm of $e$ is $1$ .

$\frac{1}{2} (\ln (5) + 3 \cdot 1 + 4 \ln (x))$

Step 8

Multiply $3$ by $1$ .

$\frac{1}{2} (\ln (5) + 3 + 4 \ln (x))$

Step 9

Apply the distributive property.

$\frac{1}{2} \ln (5) + \frac{1}{2} \cdot 3 + \frac{1}{2} (4 \ln (x))$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{2}$ and $\ln (5)$ .

$\frac{\ln (5)}{2} + \frac{1}{2} \cdot 3 + \frac{1}{2} (4 \ln (x))$

Step 10.2

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

$\frac{\ln (5)}{2} + \frac{3}{2} + \frac{1}{2} (4 \ln (x))$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Factor $2$ out of $4 \ln (x)$ .

$\frac{\ln (5)}{2} + \frac{3}{2} + \frac{1}{2} (2 (2 \ln (x)))$

Step 10.3.2

Cancel the common factor.

$\frac{\ln (5)}{2} + \frac{3}{2} + \frac{1}{2} (2 (2 \ln (x)))$

Step 10.3.3

Rewrite the expression.

$\frac{\ln (5)}{2} + \frac{3}{2} + 2 \ln (x)$