Precalculus Examples

$\log (x) - \log (x + 1) = - 1$

Step 1

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

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

Step 2

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

$10^{- 1} = \frac{x}{x + 1}$

Step 3

Cross multiply to remove the fraction.

$x = 10^{- 1} (x + 1)$

Step 4

Simplify $10^{- 1} (x + 1)$ .

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Step 4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{10} (x + 1)$

Step 4.2

Apply the distributive property.

$x = \frac{1}{10} x + \frac{1}{10} \cdot 1$

Step 4.3

Combine $\frac{1}{10}$ and $x$ .

$x = \frac{x}{10} + \frac{1}{10} \cdot 1$

Step 4.4

Multiply $\frac{1}{10}$ by $1$ .

$x = \frac{x}{10} + \frac{1}{10}$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $\frac{x}{10}$ from both sides of the equation.

$x - \frac{x}{10} = \frac{1}{10}$

Step 5.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x \cdot \frac{10}{10} - \frac{x}{10} = \frac{1}{10}$

Step 5.3

Combine $x$ and $\frac{10}{10}$ .

$\frac{x \cdot 10}{10} - \frac{x}{10} = \frac{1}{10}$

Step 5.4

Combine the numerators over the common denominator.

$\frac{x \cdot 10 - x}{10} = \frac{1}{10}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Move $10$ to the left of $x$ .

$\frac{10 \cdot x - x}{10} = \frac{1}{10}$

Step 5.5.2

Subtract $x$ from $10 x$ .

$\frac{9 x}{10} = \frac{1}{10}$

Step 6

Multiply both sides by $10$ .

$\frac{9 x}{10} \cdot 10 = \frac{1}{10} \cdot 10$

Step 7

Simplify.

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Step 7.1

Simplify the left side.

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Step 7.1.1

Cancel the common factor of $10$ .

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Step 7.1.1.1

Cancel the common factor.

$\frac{9 x}{10} \cdot 10 = \frac{1}{10} \cdot 10$

Step 7.1.1.2

Rewrite the expression.

$9 x = \frac{1}{10} \cdot 10$

Step 7.2

Simplify the right side.

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Step 7.2.1

Cancel the common factor of $10$ .

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Step 7.2.1.1

Cancel the common factor.

$9 x = \frac{1}{10} \cdot 10$

Step 7.2.1.2

Rewrite the expression.

$9 x = 1$

Step 8

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 8.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $9$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{9}$

Decimal Form:

$x = 0.\overline{1}$