Precalculus Examples

$\frac{1}{2} \cdot \log_{2} (x) = 2 \log_{2} (4)$

Step 1

Multiply both sides of the equation by $2$ .

$2 (\frac{1}{2} \cdot \log_{2} (x)) = 2 (2 \log_{2} (4))$

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

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Step 2.1.1.1

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

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

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.

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

Step 2.1.1.2.2

Rewrite the expression.

$\log_{2} (x) = 2 (2 \log_{2} (4))$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $2 (2 \log_{2} (4))$ .

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Step 2.2.1.1

Logarithm base $2$ of $4$ is $2$ .

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

Step 2.2.1.2

Multiply $2 (2 \cdot 2)$ .

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Step 2.2.1.2.1

Multiply $2$ by $2$ .

$\log_{2} (x) = 2 \cdot 4$

Step 2.2.1.2.2

Multiply $2$ by $4$ .

$\log_{2} (x) = 8$

Step 3

Rewrite $\log_{2} (x) = 8$ 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 $b^{y} = x$ .

$2^{8} = x$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x = 2^{8}$ .

$x = 2^{8}$

Step 4.2

Raise $2$ to the power of $8$ .

$x = 256$