Precalculus Examples

$\log_{5} (4 x + 12) - \log_{5} (x - 13) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{5} (\frac{4 x + 12}{x - 13}) = 3$

Step 1.2

Factor $4$ out of $4 x + 12$ .

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Step 1.2.1

Factor $4$ out of $4 x$ .

$\log_{5} (\frac{4 (x) + 12}{x - 13}) = 3$

Step 1.2.2

Factor $4$ out of $12$ .

$\log_{5} (\frac{4 x + 4 \cdot 3}{x - 13}) = 3$

Step 1.2.3

Factor $4$ out of $4 x + 4 \cdot 3$ .

$\log_{5} (\frac{4 (x + 3)}{x - 13}) = 3$

Step 2

Rewrite $\log_{5} (\frac{4 (x + 3)}{x - 13}) = 3$ 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 $b^{y} = x$ .

$5^{3} = \frac{4 (x + 3)}{x - 13}$

Step 3

Cross multiply to remove the fraction.

$4 (x + 3) = 5^{3} (x - 13)$

Step 4

Simplify $5^{3} (x - 13)$ .

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Step 4.1

Raise $5$ to the power of $3$ .

$4 (x + 3) = 125 (x - 13)$

Step 4.2

Apply the distributive property.

$4 (x + 3) = 125 x + 125 \cdot - 13$

Step 4.3

Multiply $125$ by $- 13$ .

$4 (x + 3) = 125 x - 1625$

Step 5

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

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Step 5.1

Subtract $125 x$ from both sides of the equation.

$4 (x + 3) - 125 x = - 1625$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$4 x + 4 \cdot 3 - 125 x = - 1625$

Step 5.2.2

Multiply $4$ by $3$ .

$4 x + 12 - 125 x = - 1625$

Step 5.3

Subtract $125 x$ from $4 x$ .

$- 121 x + 12 = - 1625$

Step 6

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

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Step 6.1

Subtract $12$ from both sides of the equation.

$- 121 x = - 1625 - 12$

Step 6.2

Subtract $12$ from $- 1625$ .

$- 121 x = - 1637$

Step 7

Divide each term in $- 121 x = - 1637$ by $- 121$ and simplify.

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Step 7.1

Divide each term in $- 121 x = - 1637$ by $- 121$ .

$\frac{- 121 x}{- 121} = \frac{- 1637}{- 121}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 121$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 121 x}{- 121} = \frac{- 1637}{- 121}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1637}{- 121}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{1637}{121}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1637}{121}$

Decimal Form:

$x = 13.52892561 \dots$

Mixed Number Form:

$x = 13 \frac{64}{121}$