Algebra Examples

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

Step 1

Multiply both sides of the equation by $\frac{3}{2}$ .

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

Step 2

Simplify both sides of the equation.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $\frac{3}{2} (\frac{2}{3} \cdot \log_{3} (x))$ .

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Step 2.1.1.1

Combine $\frac{2}{3}$ and $\log_{3} (x)$ .

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

Step 2.1.1.2

Cancel the common factor of $3$ .

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Step 2.1.1.2.1

Cancel the common factor.

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

Step 2.1.1.2.2

Rewrite the expression.

$\frac{1}{2} (2 \log_{3} (x)) = \frac{3}{2} \cdot 2$

Step 2.1.1.3

Cancel the common factor of $2$ .

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Step 2.1.1.3.1

Factor $2$ out of $2 \log_{3} (x)$ .

$\frac{1}{2} (2 (\log_{3} (x))) = \frac{3}{2} \cdot 2$

Step 2.1.1.3.2

Cancel the common factor.

$\frac{1}{2} (2 \log_{3} (x)) = \frac{3}{2} \cdot 2$

Step 2.1.1.3.3

Rewrite the expression.

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 3

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

$3^{3} = x$

Step 4

Solve for $x$ .

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Step 4.1

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

$x = 3^{3}$

Step 4.2

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

$x = 27$