Precalculus Examples

$\ln (\frac{9}{8 x^{8} y})$

Step 1

Rewrite $\ln (\frac{9}{8 x^{8} y})$ as $\ln (9) - \ln (8 x^{8} y)$ .

$\ln (9) - \ln (8 x^{8} y)$

Step 2

Rewrite $\ln (8 x^{8} y)$ as $\ln (8 x^{8}) + \ln (y)$ .

$\ln (9) - (\ln (8 x^{8}) + \ln (y))$

Step 3

Rewrite $\ln (8 x^{8})$ as $\ln (8) + \ln (x^{8})$ .

$\ln (9) - (\ln (8) + \ln (x^{8}) + \ln (y))$

Step 4

Expand $\ln (x^{8})$ by moving $8$ outside the logarithm.

$\ln (9) - (\ln (8) + 8 \ln (x) + \ln (y))$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

$\ln (3^{2}) - (\ln (8) + 8 \ln (x) + \ln (y))$

Step 5.2

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

$2 \ln (3) - (\ln (8) + 8 \ln (x) + \ln (y))$

Step 5.3

Simplify each term.

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Step 5.3.1

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

$2 \ln (3) - (\ln (2^{3}) + 8 \ln (x) + \ln (y))$

Step 5.3.2

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

$2 \ln (3) - (3 \ln (2) + 8 \ln (x) + \ln (y))$

Step 5.4

Apply the distributive property.

$2 \ln (3) - (3 \ln (2)) - (8 \ln (x)) - \ln (y)$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $3$ by $- 1$ .

$2 \ln (3) - 3 \ln (2) - (8 \ln (x)) - \ln (y)$

Step 5.5.2

Multiply $8$ by $- 1$ .

$2 \ln (3) - 3 \ln (2) - 8 \ln (x) - \ln (y)$