Precalculus Examples

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

Step 1

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

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

Step 2

Rewrite $\ln (\frac{x - 4}{x}) = 5$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{5} = \frac{x - 4}{x}$

Step 3

Cross multiply to remove the fraction.

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

Step 4

Multiply $e^{5}$ by $x$ .

$x - 4 = e^{5} x$

Step 5

Subtract $e^{5} x$ from both sides of the equation.

$x - 4 - e^{5} x = 0$

Step 6

Add $4$ to both sides of the equation.

$x - e^{5} x = 4$

Step 7

Factor $x$ out of $x - e^{5} x$ .

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Step 7.1

Factor $x$ out of $x^{1}$ .

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

Step 7.2

Factor $x$ out of $- e^{5} x$ .

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

Step 7.3

Factor $x$ out of $x \cdot 1 + x (- e^{5})$ .

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

Step 8

Divide each term in $x (1 - e^{5}) = 4$ by $1 - e^{5}$ and simplify.

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Step 8.1

Divide each term in $x (1 - e^{5}) = 4$ by $1 - e^{5}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $1 - e^{5}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{1 - e^{5}}$

Step 9

Exclude the solutions that do not make $\ln (x - 4) - \ln (x) = 5$ true.

$No solution$