Precalculus Examples

$\log_{12} (2 x - 1) - \log_{12} (x - 2) = 1$

Step 1

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

$\log_{12} (\frac{2 x - 1}{x - 2}) = 1$

Step 2

Rewrite $\log_{12} (\frac{2 x - 1}{x - 2}) = 1$ 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 $b^{y} = x$ .

$12^{1} = \frac{2 x - 1}{x - 2}$

Step 3

Cross multiply to remove the fraction.

$2 x - 1 = 12 (x - 2)$

Step 4

Simplify $12 (x - 2)$ .

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Step 4.1

Apply the distributive property.

$2 x - 1 = 12 x + 12 \cdot - 2$

Step 4.2

Multiply $12$ by $- 2$ .

$2 x - 1 = 12 x - 24$

Step 5

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

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Step 5.1

Subtract $12 x$ from both sides of the equation.

$2 x - 1 - 12 x = - 24$

Step 5.2

Subtract $12 x$ from $2 x$ .

$- 10 x - 1 = - 24$

Step 6

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

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Step 6.1

Add $1$ to both sides of the equation.

$- 10 x = - 24 + 1$

Step 6.2

Add $- 24$ and $1$ .

$- 10 x = - 23$

Step 7

Divide each term in $- 10 x = - 23$ by $- 10$ and simplify.

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Step 7.1

Divide each term in $- 10 x = - 23$ by $- 10$ .

$\frac{- 10 x}{- 10} = \frac{- 23}{- 10}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 10$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 10 x}{- 10} = \frac{- 23}{- 10}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 23}{- 10}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Dividing two negative values results in a positive value.

$x = \frac{23}{10}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{23}{10}$

Decimal Form:

$x = 2.3$

Mixed Number Form:

$x = 2 \frac{3}{10}$