Precalculus Examples

$\log_{3} (4 x - 9) = 2$

Step 1

Rewrite $\log_{3} (4 x - 9) = 2$ 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 $b^{y} = x$ .

$3^{2} = 4 x - 9$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $4 x - 9 = 3^{2}$ .

$4 x - 9 = 3^{2}$

Step 2.2

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

$4 x - 9 = 9$

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 $9$ to both sides of the equation.

$4 x = 9 + 9$

Step 2.3.2

Add $9$ and $9$ .

$4 x = 18$

Step 2.4

Divide each term in $4 x = 18$ by $4$ and simplify.

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Step 2.4.1

Divide each term in $4 x = 18$ by $4$ .

$\frac{4 x}{4} = \frac{18}{4}$

Step 2.4.2

Simplify the left side.

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Step 2.4.2.1

Cancel the common factor of $4$ .

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Step 2.4.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{18}{4}$

Step 2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{18}{4}$

Step 2.4.3

Simplify the right side.

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Step 2.4.3.1

Cancel the common factor of $18$ and $4$ .

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Step 2.4.3.1.1

Factor $2$ out of $18$ .

$x = \frac{2 (9)}{4}$

Step 2.4.3.1.2

Cancel the common factors.

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Step 2.4.3.1.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 9}{2 \cdot 2}$

Step 2.4.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 9}{2 \cdot 2}$

Step 2.4.3.1.2.3

Rewrite the expression.

$x = \frac{9}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{2}$

Decimal Form:

$x = 4.5$

Mixed Number Form:

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