Finite Math Examples

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

Step 1

Simplify the left side.

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Step 1.1

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

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

Step 1.2

Simplify the numerator.

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Step 1.2.1

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

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

Step 1.2.2

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

$\log (\frac{(x + 1) (x - 1)}{x - 1}) = 1$

Step 1.3

Cancel the common factor of $x - 1$ .

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Step 1.3.1

Cancel the common factor.

$\log (\frac{(x + 1) (x - 1)}{x - 1}) = 1$

Step 1.3.2

Divide $x + 1$ by $1$ .

$\log (x + 1) = 1$

Step 2

Rewrite $\log (x + 1) = 1$ 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 $b^{y} = x$ .

$10^{1} = x + 1$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x + 1 = 10^{1}$ .

$x + 1 = 10$

Step 3.2

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

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Step 3.2.1

Subtract $1$ from both sides of the equation.

$x = 10 - 1$

Step 3.2.2

Subtract $1$ from $10$ .

$x = 9$