Finite Math Examples

$\log (7 x + 4) - \log (x - 5) = 1$

Step 1

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

$\log (\frac{7 x + 4}{x - 5}) = 1$

Step 2

Rewrite $\log (\frac{7 x + 4}{x - 5}) = 1$ 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 $b^{y} = x$ .

$10^{1} = \frac{7 x + 4}{x - 5}$

Step 3

Cross multiply to remove the fraction.

$7 x + 4 = 10 (x - 5)$

Step 4

Simplify $10 (x - 5)$ .

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Step 4.1

Apply the distributive property.

$7 x + 4 = 10 x + 10 \cdot - 5$

Step 4.2

Multiply $10$ by $- 5$ .

$7 x + 4 = 10 x - 50$

Step 5

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

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Step 5.1

Subtract $10 x$ from both sides of the equation.

$7 x + 4 - 10 x = - 50$

Step 5.2

Subtract $10 x$ from $7 x$ .

$- 3 x + 4 = - 50$

Step 6

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

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Step 6.1

Subtract $4$ from both sides of the equation.

$- 3 x = - 50 - 4$

Step 6.2

Subtract $4$ from $- 50$ .

$- 3 x = - 54$

Step 7

Divide each term in $- 3 x = - 54$ by $- 3$ and simplify.

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Step 7.1

Divide each term in $- 3 x = - 54$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 54}{- 3}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 3$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 54}{- 3}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 54}{- 3}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 54$ by $- 3$ .

$x = 18$