Finite Math Examples

$\log (x - 6) = \log (- x + 14)$

Step 1

Subtract $\log (- x + 14)$ from both sides of the equation.

$\log (x - 6) - \log (- x + 14) = 0$

Step 2

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

$\log (\frac{x - 6}{- x + 14}) = 0$

Step 3

Rewrite $\log (\frac{x - 6}{- x + 14}) = 0$ 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 $b^{y} = x$ .

$10^{0} = \frac{x - 6}{- x + 14}$

Step 4

Cross multiply to remove the fraction.

$x - 6 = 10^{0} (- x + 14)$

Step 5

Simplify $10^{0} (- x + 14)$ .

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Step 5.1

Remove parentheses.

$x - 6 = 10^{0} (- x + 14)$

Step 5.2

Anything raised to $0$ is $1$ .

$x - 6 = 1 (- x + 14)$

Step 5.3

Multiply $- x + 14$ by $1$ .

$x - 6 = - x + 14$

Step 6

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

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Step 6.1

Add $x$ to both sides of the equation.

$x - 6 + x = 14$

Step 6.2

Add $x$ and $x$ .

$2 x - 6 = 14$

Step 7

Factor $2$ out of $2 x - 6$ .

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Step 7.1

Factor $2$ out of $2 x$ .

$2 (x) - 6 = 14$

Step 7.2

Factor $2$ out of $- 6$ .

$2 x + 2 \cdot - 3 = 14$

Step 7.3

Factor $2$ out of $2 x + 2 \cdot - 3$ .

$2 (x - 3) = 14$

Step 8

Divide each term in $2 (x - 3) = 14$ by $2$ and simplify.

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Step 8.1

Divide each term in $2 (x - 3) = 14$ by $2$ .

$\frac{2 (x - 3)}{2} = \frac{14}{2}$

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Cancel the common factor.

$\frac{2 (x - 3)}{2} = \frac{14}{2}$

Step 8.2.1.2

Divide $x - 3$ by $1$ .

$x - 3 = \frac{14}{2}$

Step 8.3

Simplify the right side.

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Step 8.3.1

Divide $14$ by $2$ .

$x - 3 = 7$

Step 9

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

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Step 9.1

Add $3$ to both sides of the equation.

$x = 7 + 3$

Step 9.2

Add $7$ and $3$ .

$x = 10$