Precalculus Examples

$2 \ln (5 e^{3} x y^{- 1} z^{2})$

Step 1

Rewrite $\ln (5 e^{3} x y^{- 1} z^{2})$ as $\ln (5 e^{3} x y^{- 1}) + \ln (z^{2})$ .

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

Step 2

Rewrite $\ln (5 e^{3} x y^{- 1})$ as $\ln (5 e^{3} x) + \ln (y^{- 1})$ .

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

Step 3

Rewrite $\ln (5 e^{3} x)$ as $\ln (5 e^{3}) + \ln (x)$ .

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

Step 4

Rewrite $\ln (5 e^{3})$ as $\ln (5) + \ln (e^{3})$ .

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

Step 5

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

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

Step 6

Expand $\ln (y^{- 1})$ by moving $- 1$ outside the logarithm.

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

Step 7

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

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

Step 8

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

$2 (\ln (5) + 3 \cdot 1 + \ln (x) - \ln (y) + 2 \ln (z))$

Step 9

Multiply $3$ by $1$ .

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

Step 10

Apply the distributive property.

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

Step 11

Simplify.

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Step 11.1

Multiply $2$ by $3$ .

$2 \ln (5) + 6 + 2 \ln (x) + 2 (- \ln (y)) + 2 (2 \ln (z))$

Step 11.2

Multiply $- 1$ by $2$ .

$2 \ln (5) + 6 + 2 \ln (x) - 2 \ln (y) + 2 (2 \ln (z))$

Step 11.3

Multiply $2$ by $2$ .

$2 \ln (5) + 6 + 2 \ln (x) - 2 \ln (y) + 4 \ln (z)$