Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Expand $\ln ({(x - 1)}^{3})$ by moving $3$ outside the logarithm.

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

Step 5

Expand $\ln ({(x^{2} + 1)}^{- 2})$ by moving $- 2$ outside the logarithm.

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

Step 6

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

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

Step 7

Multiply $- 2$ by $1$ .

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