Precalculus Examples

$2 \log_{3} (x - 3) = 4$

Step 1

Divide each term in $2 \log_{3} (x - 3) = 4$ by $2$ and simplify.

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Step 1.1

Divide each term in $2 \log_{3} (x - 3) = 4$ by $2$ .

$\frac{2 \log_{3} (x - 3)}{2} = \frac{4}{2}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $2$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{2 \log_{3} (x - 3)}{2} = \frac{4}{2}$

Step 1.2.1.2

Divide $\log_{3} (x - 3)$ by $1$ .

$\log_{3} (x - 3) = \frac{4}{2}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $4$ by $2$ .

$\log_{3} (x - 3) = 2$

Step 2

Rewrite $\log_{3} (x - 3) = 2$ 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 $b^{y} = x$ .

$3^{2} = x - 3$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $x - 3 = 3^{2}$ .

$x - 3 = 3^{2}$

Step 3.2

Raise $3$ to the power of $2$ .

$x - 3 = 9$

Step 3.3

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

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Step 3.3.1

Add $3$ to both sides of the equation.

$x = 9 + 3$

Step 3.3.2

Add $9$ and $3$ .

$x = 12$