Precalculus Examples

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

Step 1

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

$3^{2} = 3 x - 2$

Step 2

Solve for $x$ .

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Step 2.1

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

$3 x - 2 = 3^{2}$

Step 2.2

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

$3 x - 2 = 9$

Step 2.3

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

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Step 2.3.1

Add $2$ to both sides of the equation.

$3 x = 9 + 2$

Step 2.3.2

Add $9$ and $2$ .

$3 x = 11$

Step 2.4

Divide each term in $3 x = 11$ by $3$ and simplify.

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Step 2.4.1

Divide each term in $3 x = 11$ by $3$ .

$\frac{3 x}{3} = \frac{11}{3}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{11}{3}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11}{3}$

Decimal Form:

$x = 3.\overline{6}$

Mixed Number Form:

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