Precalculus Examples

$\log (x + 1) = 2 + \log (x - 6)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log (x + 1) - \log (x - 6) = 2$

Step 2

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

$\log (\frac{x + 1}{x - 6}) = 2$

Step 3

Rewrite $\log (\frac{x + 1}{x - 6}) = 2$ 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 $b^{y} = x$ .

$10^{2} = \frac{x + 1}{x - 6}$

Step 4

Cross multiply to remove the fraction.

$x + 1 = 10^{2} (x - 6)$

Step 5

Simplify $10^{2} (x - 6)$ .

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Step 5.1

Raise $10$ to the power of $2$ .

$x + 1 = 100 (x - 6)$

Step 5.2

Apply the distributive property.

$x + 1 = 100 x + 100 \cdot - 6$

Step 5.3

Multiply $100$ by $- 6$ .

$x + 1 = 100 x - 600$

Step 6

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

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Step 6.1

Subtract $100 x$ from both sides of the equation.

$x + 1 - 100 x = - 600$

Step 6.2

Subtract $100 x$ from $x$ .

$- 99 x + 1 = - 600$

Step 7

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

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Step 7.1

Subtract $1$ from both sides of the equation.

$- 99 x = - 600 - 1$

Step 7.2

Subtract $1$ from $- 600$ .

$- 99 x = - 601$

Step 8

Divide each term in $- 99 x = - 601$ by $- 99$ and simplify.

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Step 8.1

Divide each term in $- 99 x = - 601$ by $- 99$ .

$\frac{- 99 x}{- 99} = \frac{- 601}{- 99}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $- 99$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{- 99 x}{- 99} = \frac{- 601}{- 99}$

Step 8.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 601}{- 99}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Dividing two negative values results in a positive value.

$x = \frac{601}{99}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{601}{99}$

Decimal Form:

$x = 6.\overline{07}$

Mixed Number Form:

$x = 6 \frac{7}{99}$