Algebra Examples

ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})

$\ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})$

Step 1

Rewrite

ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})

$\ln (\frac{8 e^{2} x^{5}}{(x^{2} + 1) \sqrt[3]{x + 2}})$ as

ln (8 e^{2} x^{5}) - ln ((x^{2} + 1) \sqrt[3]{x + 2})

$\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) \sqrt[3]{x + 2})$ .

ln (8 e^{2} x^{5}) - ln ((x^{2} + 1) \sqrt[3]{x + 2})

$\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) \sqrt[3]{x + 2})$

Step 2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt[3]{x + 2}

$\sqrt[3]{x + 2}$ as

{(x + 2)}^{\frac{1}{3}}

${(x + 2)}^{\frac{1}{3}}$ .

ln (8 e^{2} x^{5}) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8 e^{2} x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 3

Rewrite

ln (8 e^{2} x^{5})

$\ln (8 e^{2} x^{5})$ as

ln (8 e^{2}) + ln (x^{5})

$\ln (8 e^{2}) + \ln (x^{5})$ .

ln (8 e^{2}) + ln (x^{5}) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8 e^{2}) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 4

Rewrite

ln (8 e^{2})

$\ln (8 e^{2})$ as

ln (8) + ln (e^{2})

$\ln (8) + \ln (e^{2})$ .

ln (8) + ln (e^{2}) + ln (x^{5}) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8) + \ln (e^{2}) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 5

Expand

ln (e^{2})

$\ln (e^{2})$ by moving

2

$2$ outside the logarithm.

ln (8) + 2 ln (e) + ln (x^{5}) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8) + 2 \ln (e) + \ln (x^{5}) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 6

Expand

ln (x^{5})

$\ln (x^{5})$ by moving

5

$5$ outside the logarithm.

ln (8) + 2 ln (e) + 5 ln (x) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8) + 2 \ln (e) + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 7

The natural logarithm of

e

$e$ is

1

$1$ .

ln (8) + 2 \cdot 1 + 5 ln (x) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8) + 2 \cdot 1 + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 8

Multiply

2

$2$ by

1

$1$ .

ln (8) + 2 + 5 ln (x) - ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})

$\ln (8) + 2 + 5 \ln (x) - \ln ((x^{2} + 1) {(x + 2)}^{\frac{1}{3}})$

Step 9

Rewrite

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

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

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

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

ln (8) + 2 + 5 ln (x) - (ln (x^{2} + 1) + ln ({(x + 2)}^{\frac{1}{3}}))

$\ln (8) + 2 + 5 \ln (x) - (\ln (x^{2} + 1) + \ln ({(x + 2)}^{\frac{1}{3}}))$

Step 10

Expand

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

$\ln ({(x + 2)}^{\frac{1}{3}})$ by moving

\frac{1}{3}

$\frac{1}{3}$ outside the logarithm.

ln (8) + 2 + 5 ln (x) - (ln (x^{2} + 1) + \frac{1}{3} ln (x + 2))

$\ln (8) + 2 + 5 \ln (x) - (\ln (x^{2} + 1) + \frac{1}{3} \ln (x + 2))$

Step 11

Simplify each term.

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Step 11.1

Rewrite

ln (8)

$\ln (8)$ as

ln (2^{3})

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

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

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

Step 11.2

Expand

ln (2^{3})

$\ln (2^{3})$ by moving

3

$3$ outside the logarithm.

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

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

Step 11.3

Combine

\frac{1}{3}

$\frac{1}{3}$ and

ln (x + 2)

$\ln (x + 2)$ .

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

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

Step 11.4

To write

ln (x^{2} + 1)

$\ln (x^{2} + 1)$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

3 ln (2) + 2 + 5 ln (x) - (ln (x^{2} + 1) \cdot \frac{3}{3} + \frac{ln (x + 2)}{3})

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

Step 11.5

Combine

ln (x^{2} + 1)

$\ln (x^{2} + 1)$ and

\frac{3}{3}

$\frac{3}{3}$ .

3 ln (2) + 2 + 5 ln (x) - (\frac{ln (x^{2} + 1) \cdot 3}{3} + \frac{ln (x + 2)}{3})

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

Step 11.6

Combine the numerators over the common denominator.

3 ln (2) + 2 + 5 ln (x) - \frac{ln (x^{2} + 1) \cdot 3 + ln (x + 2)}{3}

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

Step 11.7

Move

$3$ to the left of

$\ln (x^{2} + 1)$ .

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