Precalculus Examples

$\ln (\frac{x^{2} \sqrt{p q^{9}}}{e^{4}})$

Step 1

Rewrite $\frac{x^{2} \sqrt{p q^{9}}}{e^{4}}$ as $x^{2} \sqrt{p q^{9}} {(e^{4})}^{- 1}$ .

$\ln (x^{2} \sqrt{p q^{9}} {(e^{4})}^{- 1})$

Step 2

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

$\ln (x^{2} \sqrt{p q^{9}} e^{4 \cdot - 1})$

Step 3

Rewrite $\ln (x^{2} \sqrt{p q^{9}} e^{4 \cdot - 1})$ as $\ln (x^{2} \sqrt{p q^{9}}) + \ln (e^{4 \cdot - 1})$ .

$\ln (x^{2} \sqrt{p q^{9}}) + \ln (e^{4 \cdot - 1})$

Step 4

Use logarithm rules to move $4 \cdot - 1$ out of the exponent.

$\ln (x^{2} \sqrt{p q^{9}}) + 4 \cdot - 1 \ln (e)$

Step 5

Multiply $4$ by $- 1$ .

$\ln (x^{2} \sqrt{p q^{9}}) - 4 \ln (e)$

Step 6

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

$\ln (x^{2} \sqrt{p q^{9}}) - 4 \cdot 1$

Step 7

Multiply $- 4$ by $1$ .

$\ln (x^{2} \sqrt{p q^{9}}) - 4$

Step 8

Simplify each term.

Tap for more steps...

Step 8.1

Rewrite $p q^{9}$ as ${({(q^{2})}^{2})}^{2} (p q)$ .

Tap for more steps...

Step 8.1.1

Factor out $q^{8}$ .

$\ln (x^{2} \sqrt{p (q^{8} q)}) - 4$

Step 8.1.2

Rewrite $q^{8}$ as ${(q^{4})}^{2}$ .

$\ln (x^{2} \sqrt{p ({(q^{4})}^{2} q)}) - 4$

Step 8.1.3

Reorder $p$ and ${(q^{4})}^{2}$ .

$\ln (x^{2} \sqrt{{(q^{4})}^{2} p q}) - 4$

Step 8.1.4

Rewrite $q^{4}$ as ${(q^{2})}^{2}$ .

$\ln (x^{2} \sqrt{{({(q^{2})}^{2})}^{2} p q}) - 4$

Step 8.1.5

Add parentheses.

$\ln (x^{2} \sqrt{{({(q^{2})}^{2})}^{2} (p q)}) - 4$

Step 8.2

Pull terms out from under the radical.

$\ln (x^{2} ({(q^{2})}^{2} \sqrt{p q})) - 4$

Step 8.3

Multiply the exponents in ${(q^{2})}^{2}$ .

Tap for more steps...

Step 8.3.1

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

$\ln (x^{2} (q^{2 \cdot 2} \sqrt{p q})) - 4$

Step 8.3.2

Multiply $2$ by $2$ .

$\ln (x^{2} (q^{4} \sqrt{p q})) - 4$

$\ln (x^{2} q^{4} \sqrt{p q}) - 4$

Step 9

Reorder $x^{2}$ and $q^{4}$ .

$\ln (q^{4} x^{2} \sqrt{p q}) - 4$