Finite Math Examples

$\ln (x - 4) = \ln (x + 8) - \ln (5)$

Step 1

Simplify the right side.

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Step 1.1

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

$\ln (x - 4) = \ln (\frac{x + 8}{5})$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x - 4 = \frac{x + 8}{5}$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides by $5$ .

$(x - 4) \cdot 5 = \frac{x + 8}{5} \cdot 5$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $(x - 4) \cdot 5$ .

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Step 3.2.1.1.1

Apply the distributive property.

$x \cdot 5 - 4 \cdot 5 = \frac{x + 8}{5} \cdot 5$

Step 3.2.1.1.2

Simplify the expression.

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Step 3.2.1.1.2.1

Move $5$ to the left of $x$ .

$5 \cdot x - 4 \cdot 5 = \frac{x + 8}{5} \cdot 5$

Step 3.2.1.1.2.2

Multiply $- 4$ by $5$ .

$5 x - 20 = \frac{x + 8}{5} \cdot 5$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$5 x - 20 = \frac{x + 8}{5} \cdot 5$

Step 3.2.2.1.2

Rewrite the expression.

$5 x - 20 = x + 8$

Step 3.3

Solve for $x$ .

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $x$ from both sides of the equation.

$5 x - 20 - x = 8$

Step 3.3.1.2

Subtract $x$ from $5 x$ .

$4 x - 20 = 8$

Step 3.3.2

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

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Step 3.3.2.1

Add $20$ to both sides of the equation.

$4 x = 8 + 20$

Step 3.3.2.2

Add $8$ and $20$ .

$4 x = 28$

Step 3.3.3

Divide each term in $4 x = 28$ by $4$ and simplify.

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Step 3.3.3.1

Divide each term in $4 x = 28$ by $4$ .

$\frac{4 x}{4} = \frac{28}{4}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $4$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{28}{4}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{28}{4}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $28$ by $4$ .

$x = 7$