Algebra Examples

$\frac{1}{3} \cdot \log_{5} (12 x) = 2$

Step 1

Multiply both sides of the equation by $3$ .

$3 (\frac{1}{3} \cdot \log_{5} (12 x)) = 3 \cdot 2$

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 $3 (\frac{1}{3} \cdot \log_{5} (12 x))$ .

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Step 2.1.1.1

Combine $\frac{1}{3}$ and $\log_{5} (12 x)$ .

$3 \frac{\log_{5} (12 x)}{3} = 3 \cdot 2$

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.

$3 \frac{\log_{5} (12 x)}{3} = 3 \cdot 2$

Step 2.1.1.2.2

Rewrite the expression.

$\log_{5} (12 x) = 3 \cdot 2$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $3$ by $2$ .

$\log_{5} (12 x) = 6$

Step 3

Rewrite $\log_{5} (12 x) = 6$ 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 $b^{y} = x$ .

$5^{6} = 12 x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $12 x = 5^{6}$ .

$12 x = 5^{6}$

Step 4.2

Divide each term in $12 x = 5^{6}$ by $12$ and simplify.

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Step 4.2.1

Divide each term in $12 x = 5^{6}$ by $12$ .

$\frac{12 x}{12} = \frac{5^{6}}{12}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $12$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{5^{6}}{12}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5^{6}}{12}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Raise $5$ to the power of $6$ .

$x = \frac{15625}{12}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{15625}{12}$

Decimal Form:

$x = 1302.08\overline{3}$

Mixed Number Form:

$x = 1302 \frac{1}{12}$