Precalculus Examples

$\log_{9} (x + 6) - \log_{9} (x) = \log_{9} (2)$

Step 1

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

$\log_{9} (\frac{x + 6}{x}) = \log_{9} (2)$

Step 2

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

$\frac{x + 6}{x} = 2$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$x$

Step 3.2

Multiply each term in $\frac{x + 6}{x} = 2$ by $x$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{x + 6}{x} = 2$ by $x$ .

$\frac{x + 6}{x} x = 2 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{x + 6}{x} x = 2 x$

Step 3.2.2.1.2

Rewrite the expression.

$x + 6 = 2 x$

Step 3.3

Solve the equation.

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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 $2 x$ from both sides of the equation.

$x + 6 - 2 x = 0$

Step 3.3.1.2

Subtract $2 x$ from $x$ .

$- x + 6 = 0$

Step 3.3.2

Subtract $6$ from both sides of the equation.

$- x = - 6$

Step 3.3.3

Divide each term in $- x = - 6$ by $- 1$ and simplify.

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Step 3.3.3.1

Divide each term in $- x = - 6$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 6}{- 1}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 6}{- 1}$

Step 3.3.3.2.2

Divide $x$ by $1$ .

$x = \frac{- 6}{- 1}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $- 6$ by $- 1$ .

$x = 6$