Precalculus Examples

$\ln (\frac{e^{x}}{5 e^{- x}})$

Step 1

Rewrite $\ln (\frac{e^{x}}{5 e^{- x}})$ as $\ln (e^{x}) - \ln (5 e^{- x})$ .

$\ln (e^{x}) - \ln (5 e^{- x})$

Step 2

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

$x \ln (e) - \ln (5 e^{- x})$

Step 3

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

$x \cdot 1 - \ln (5 e^{- x})$

Step 4

Multiply $x$ by $1$ .

$x - \ln (5 e^{- x})$

Step 5

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

$x - (\ln (5) + \ln (e^{- x}))$

Step 6

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

$x - (\ln (5) - x \ln (e))$

Step 7

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

$x - (\ln (5) - x \cdot 1)$

Step 8

Multiply $- 1$ by $1$ .

$x - (\ln (5) - x)$

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

$x - \ln (5) - - x$

Step 9.2

Multiply $- - x$ .

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Step 9.2.1

Multiply $- 1$ by $- 1$ .

$x - \ln (5) + 1 x$

Step 9.2.2

Multiply $x$ by $1$ .

$x - \ln (5) + x$

Step 10

Add $x$ and $x$ .

$2 x - \ln (5)$