Precalculus Examples

$\frac{1}{2} \cdot \log_{3} (x) = 3 \log_{3} (11)$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot \log_{3} (x)) = 2 (3 \log_{3} (11))$

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

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Step 2.1.1.1

Combine $\frac{1}{2}$ and $\log_{3} (x)$ .

$2 \frac{\log_{3} (x)}{2} = 2 (3 \log_{3} (11))$

Step 2.1.1.2

Cancel the common factor of $2$ .

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Step 2.1.1.2.1

Cancel the common factor.

$2 \frac{\log_{3} (x)}{2} = 2 (3 \log_{3} (11))$

Step 2.1.1.2.2

Rewrite the expression.

$\log_{3} (x) = 2 (3 \log_{3} (11))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $3$ by $2$ .

$\log_{3} (x) = 6 \log_{3} (11)$

Step 3

Simplify $6 \log_{3} (11)$ by moving $6$ inside the logarithm.

$\log_{3} (x) = \log_{3} (11^{6})$

Step 4

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x = 11^{6}$

Step 5

Raise $11$ to the power of $6$ .

$x = 1771561$