Precalculus Examples

$\log_{a} (5 x - 2) - \log_{a} (3 x + 4) = 0$

Step 1

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

$\log_{a} (\frac{5 x - 2}{3 x + 4}) = 0$

Step 2

Rewrite $\log_{a} (\frac{5 x - 2}{3 x + 4}) = 0$ 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 $b^{y} = x$ .

$a^{0} = \frac{5 x - 2}{3 x + 4}$

Step 3

Cross multiply to remove the fraction.

$5 x - 2 = a^{0} (3 x + 4)$

Step 4

Simplify $a^{0} (3 x + 4)$ .

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Step 4.1

Remove parentheses.

$5 x - 2 = a^{0} (3 x + 4)$

Step 4.2

Anything raised to $0$ is $1$ .

$5 x - 2 = 1 (3 x + 4)$

Step 4.3

Multiply $3 x + 4$ by $1$ .

$5 x - 2 = 3 x + 4$

Step 5

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

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Step 5.1

Subtract $3 x$ from both sides of the equation.

$5 x - 2 - 3 x = 4$

Step 5.2

Subtract $3 x$ from $5 x$ .

$2 x - 2 = 4$

Step 6

Factor $2$ out of $2 x - 2$ .

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Step 6.1

Factor $2$ out of $2 x$ .

$2 (x) - 2 = 4$

Step 6.2

Factor $2$ out of $- 2$ .

$2 (x) + 2 (- 1) = 4$

Step 6.3

Factor $2$ out of $2 (x) + 2 (- 1)$ .

$2 (x - 1) = 4$

Step 7

Divide each term in $2 (x - 1) = 4$ by $2$ and simplify.

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Step 7.1

Divide each term in $2 (x - 1) = 4$ by $2$ .

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x - 1$ by $1$ .

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

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $4$ by $2$ .

$x - 1 = 2$

Step 8

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

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Step 8.1

Add $1$ to both sides of the equation.

$x = 2 + 1$

Step 8.2

Add $2$ and $1$ .

$x = 3$