Precalculus Examples

$\ln (e^{x}) - \ln (e^{4}) = \ln (e^{5})$

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{e^{x}}{e^{4}}) = \ln (e^{5})$

Step 1.2

Rewrite $\frac{e^{x}}{e^{4}}$ as $e^{x} {(e^{4})}^{- 1}$ .

$\ln (e^{x} {(e^{4})}^{- 1}) = \ln (e^{5})$

Step 1.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\ln (e^{x} e^{4 \cdot - 1}) = \ln (e^{5})$

Step 1.4

Rewrite $\ln (e^{x} e^{4 \cdot - 1})$ as $\ln (e^{x}) + \ln (e^{4 \cdot - 1})$ .

$\ln (e^{x}) + \ln (e^{4 \cdot - 1}) = \ln (e^{5})$

Step 1.5

Use logarithm rules to move $x$ out of the exponent.

$x \ln (e) + \ln (e^{4 \cdot - 1}) = \ln (e^{5})$

Step 1.6

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

$x \cdot 1 + \ln (e^{4 \cdot - 1}) = \ln (e^{5})$

Step 1.7

Multiply $x$ by $1$ .

$x + \ln (e^{4 \cdot - 1}) = \ln (e^{5})$

Step 1.8

Simplify each term.

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Step 1.8.1

Use logarithm rules to move $4 \cdot - 1$ out of the exponent.

$x + 4 \cdot (- 1 \ln (e)) = \ln (e^{5})$

Step 1.8.2

Multiply $4$ by $- 1$ .

$x - 4 \ln (e) = \ln (e^{5})$

Step 1.8.3

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

$x - 4 \cdot 1 = \ln (e^{5})$

Step 1.8.4

Multiply $- 4$ by $1$ .

$x - 4 = \ln (e^{5})$

Step 2

Simplify the right side.

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Step 2.1

Use logarithm rules to move $5$ out of the exponent.

$x - 4 = 5 \ln (e)$

Step 2.2

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

$x - 4 = 5 \cdot 1$

Step 2.3

Multiply $5$ by $1$ .

$x - 4 = 5$

Step 3

Move all terms not containing $x$ to the right side of the equation.

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Step 3.1

Add $4$ to both sides of the equation.

$x = 5 + 4$

Step 3.2

Add $5$ and $4$ .

$x = 9$