Precalculus Examples

$\ln (x) - \ln (x + 1) = 2$

Step 1

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

$\ln (\frac{x}{x + 1}) = 2$

Step 2

Rewrite $\ln (\frac{x}{x + 1}) = 2$ 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 $b^{y} = x$ .

$e^{2} = \frac{x}{x + 1}$

Step 3

Cross multiply to remove the fraction.

$x = e^{2} (x + 1)$

Step 4

Simplify $e^{2} (x + 1)$ .

Tap for more steps...

Apply the distributive property.

$x = e^{2} x + e^{2} \cdot 1$

Multiply $e^{2}$ by $1$ .

$x = e^{2} x + e^{2}$

Step 5

Subtract $e^{2} x$ from both sides of the equation.

$x - e^{2} x = e^{2}$

Step 6

Factor the left side of the equation.

Tap for more steps...

Factor $x$ out of $x - e^{2} x$ .

Tap for more steps...

Raise $x$ to the power of $1$ .

$x - e^{2} x = e^{2}$

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

$x \cdot 1 - e^{2} x = e^{2}$

Factor $x$ out of $- e^{2} x$ .

$x \cdot 1 + x (- e^{2}) = e^{2}$

Factor $x$ out of $x \cdot 1 + x (- e^{2})$ .

$x (1 - e^{2}) = e^{2}$

Rewrite $1$ as $1^{2}$ .

$x (1^{2} - e^{2}) = e^{2}$

Factor.

Tap for more steps...

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

$x ((1 + e) (1 - e)) = e^{2}$

Remove unnecessary parentheses.

$x (1 + e) (1 - e) = e^{2}$

Step 7

Divide each term in $x (1 + e) (1 - e) = e^{2}$ by $1 - e^{2}$ and simplify.

Tap for more steps...

Divide each term in $x (1 + e) (1 - e) = e^{2}$ by $1 - e^{2}$ .

$\frac{x (1 + e) (1 - e)}{1 - e^{2}} = \frac{e^{2}}{1 - e^{2}}$

Simplify the left side.

Tap for more steps...

Simplify the denominator.

Tap for more steps...

Rewrite $1$ as $1^{2}$ .

$\frac{x (1 + e) (1 - e)}{1^{2} - e^{2}} = \frac{e^{2}}{1 - e^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

$\frac{x (1 + e) (1 - e)}{(1 + e) (1 - e)} = \frac{e^{2}}{1 - e^{2}}$

Reduce the expression by cancelling the common factors.

Tap for more steps...

Cancel the common factor of $1 + e$ .

Tap for more steps...

Cancel the common factor.

$\frac{x (1 + e) (1 - e)}{(1 + e) (1 - e)} = \frac{e^{2}}{1 - e^{2}}$

Rewrite the expression.

$\frac{x (1 - e)}{1 - e} = \frac{e^{2}}{1 - e^{2}}$

Cancel the common factor of $1 - e$ .

Tap for more steps...

Cancel the common factor.

$\frac{x (1 - e)}{1 - e} = \frac{e^{2}}{1 - e^{2}}$

Divide $x$ by $1$ .

$x = \frac{e^{2}}{1 - e^{2}}$

Simplify the right side.

Tap for more steps...

Simplify the denominator.

Tap for more steps...

Rewrite $1$ as $1^{2}$ .

$x = \frac{e^{2}}{1^{2} - e^{2}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = e$ .

$x = \frac{e^{2}}{(1 + e) (1 - e)}$

Step 8

Exclude the solutions that do not make $\ln (x) - \ln (x + 1) = 2$ true.

$No solution$