Precalculus Examples

$\ln (\frac{x^{5} y^{2}}{z^{3} w^{- 7}})$

Step 1

Rewrite $\ln (\frac{x^{5} y^{2}}{z^{3} w^{- 7}})$ as $\ln (x^{5} y^{2}) - \ln (z^{3} w^{- 7})$ .

$\ln (x^{5} y^{2}) - \ln (z^{3} w^{- 7})$

Step 2

Rewrite $\ln (x^{5} y^{2})$ as $\ln (x^{5}) + \ln (y^{2})$ .

$\ln (x^{5}) + \ln (y^{2}) - \ln (z^{3} w^{- 7})$

Step 3

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

$5 \ln (x) + \ln (y^{2}) - \ln (z^{3} w^{- 7})$

Step 4

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

$5 \ln (x) + 2 \ln (y) - \ln (z^{3} w^{- 7})$

Step 5

Rewrite $\ln (z^{3} w^{- 7})$ as $\ln (z^{3}) + \ln (w^{- 7})$ .

$5 \ln (x) + 2 \ln (y) - (\ln (z^{3}) + \ln (w^{- 7}))$

Step 6

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

$5 \ln (x) + 2 \ln (y) - (3 \ln (z) + \ln (w^{- 7}))$

Step 7

Expand $\ln (w^{- 7})$ by moving $- 7$ outside the logarithm.

$5 \ln (x) + 2 \ln (y) - (3 \ln (z) - 7 \ln (w))$

Step 8

Simplify each term.

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Step 8.1

Apply the distributive property.

$5 \ln (x) + 2 \ln (y) - (3 \ln (z)) - (- 7 \ln (w))$

Step 8.2

Multiply $3$ by $- 1$ .

$5 \ln (x) + 2 \ln (y) - 3 \ln (z) - (- 7 \ln (w))$

Step 8.3

Multiply $- 7$ by $- 1$ .

$5 \ln (x) + 2 \ln (y) - 3 \ln (z) + 7 \ln (w)$