Precalculus Examples

$\ln (\frac{4}{9 x^{9} y})$

Step 1

Rewrite $\ln (\frac{4}{9 x^{9} y})$ as $\ln (4) - \ln (9 x^{9} y)$ .

$\ln (4) - \ln (9 x^{9} y)$

Step 2

Rewrite $\ln (9 x^{9} y)$ as $\ln (9 x^{9}) + \ln (y)$ .

$\ln (4) - (\ln (9 x^{9}) + \ln (y))$

Step 3

Rewrite $\ln (9 x^{9})$ as $\ln (9) + \ln (x^{9})$ .

$\ln (4) - (\ln (9) + \ln (x^{9}) + \ln (y))$

Step 4

Expand $\ln (x^{9})$ by moving $9$ outside the logarithm.

$\ln (4) - (\ln (9) + 9 \ln (x) + \ln (y))$

Step 5

Simplify each term.

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Step 5.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

$\ln (2^{2}) - (\ln (9) + 9 \ln (x) + \ln (y))$

Step 5.2

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

$2 \ln (2) - (\ln (9) + 9 \ln (x) + \ln (y))$

Step 5.3

Simplify each term.

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Step 5.3.1

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

$2 \ln (2) - (\ln (3^{2}) + 9 \ln (x) + \ln (y))$

Step 5.3.2

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

$2 \ln (2) - (2 \ln (3) + 9 \ln (x) + \ln (y))$

Step 5.4

Apply the distributive property.

$2 \ln (2) - (2 \ln (3)) - (9 \ln (x)) - \ln (y)$

Step 5.5

Simplify.

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Step 5.5.1

Multiply $2$ by $- 1$ .

$2 \ln (2) - 2 \ln (3) - (9 \ln (x)) - \ln (y)$

Step 5.5.2

Multiply $9$ by $- 1$ .

$2 \ln (2) - 2 \ln (3) - 9 \ln (x) - \ln (y)$