Precalculus Examples

$\log_{4} (8 x) - \log_{4} (x - 1) = 2$

Step 1

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

$\log_{4} (\frac{8 x}{x - 1}) = 2$

Step 2

Rewrite $\log_{4} (\frac{8 x}{x - 1}) = 2$ 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 $b^{y} = x$ .

$4^{2} = \frac{8 x}{x - 1}$

Step 3

Cross multiply to remove the fraction.

$8 x = 4^{2} (x - 1)$

Step 4

Simplify $4^{2} (x - 1)$ .

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Step 4.1

Raise $4$ to the power of $2$ .

$8 x = 16 (x - 1)$

Step 4.2

Apply the distributive property.

$8 x = 16 x + 16 \cdot - 1$

Step 4.3

Multiply $16$ by $- 1$ .

$8 x = 16 x - 16$

Step 5

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

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Step 5.1

Subtract $16 x$ from both sides of the equation.

$8 x - 16 x = - 16$

Step 5.2

Subtract $16 x$ from $8 x$ .

$- 8 x = - 16$

Step 6

Divide each term in $- 8 x = - 16$ by $- 8$ and simplify.

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Step 6.1

Divide each term in $- 8 x = - 16$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 16}{- 8}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $- 8$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 16}{- 8}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 16}{- 8}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 16$ by $- 8$ .

$x = 2$