Precalculus Examples

$\ln (\frac{x^{4} \sqrt{p q^{5}}}{e^{6}})$

Step 1

Rewrite $\ln (\frac{x^{4} \sqrt{p q^{5}}}{e^{6}})$ as $\ln (x^{4} \sqrt{p q^{5}}) - \ln (e^{6})$ .

$\ln (x^{4} \sqrt{p q^{5}}) - \ln (e^{6})$

Step 2

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

$\ln (x^{4} {(p q^{5})}^{\frac{1}{2}}) - \ln (e^{6})$

Step 3

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

$\ln (x^{4} {(p q^{5})}^{\frac{1}{2}}) - (6 \ln (e))$

Step 4

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

$\ln (x^{4} {(p q^{5})}^{\frac{1}{2}}) - (6 \cdot 1)$

Step 5

Multiply $6$ by $1$ .

$\ln (x^{4} {(p q^{5})}^{\frac{1}{2}}) - 1 \cdot 6$

Step 6

Multiply $- 1$ by $6$ .

$\ln (x^{4} {(p q^{5})}^{\frac{1}{2}}) - 6$

Step 7

Rewrite $\ln (x^{4} {(p q^{5})}^{\frac{1}{2}})$ as $\ln (x^{4}) + \ln ({(p q^{5})}^{\frac{1}{2}})$ .

$\ln (x^{4}) + \ln ({(p q^{5})}^{\frac{1}{2}}) - 6$

Step 8

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

$4 \ln (x) + \ln ({(p q^{5})}^{\frac{1}{2}}) - 6$

Step 9

Expand $\ln ({(p q^{5})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$4 \ln (x) + \frac{1}{2} \ln (p q^{5}) - 6$

Step 10

Rewrite $\ln (p q^{5})$ as $\ln (p) + \ln (q^{5})$ .

$4 \ln (x) + \frac{1}{2} (\ln (p) + \ln (q^{5})) - 6$

Step 11

Expand $\ln (q^{5})$ by moving $5$ outside the logarithm.

$4 \ln (x) + \frac{1}{2} (\ln (p) + 5 \ln (q)) - 6$

Step 12

Simplify each term.

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Step 12.1

Apply the distributive property.

$4 \ln (x) + \frac{1}{2} \ln (p) + \frac{1}{2} (5 \ln (q)) - 6$

Step 12.2

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

$4 \ln (x) + \frac{\ln (p)}{2} + \frac{1}{2} (5 \ln (q)) - 6$

Step 12.3

Multiply $\frac{1}{2} (5 \ln (q))$ .

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Step 12.3.1

Combine $5$ and $\frac{1}{2}$ .

$4 \ln (x) + \frac{\ln (p)}{2} + \frac{5}{2} \ln (q) - 6$

Step 12.3.2

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

$4 \ln (x) + \frac{\ln (p)}{2} + \frac{5 \ln (q)}{2} - 6$