Algebra Examples

$\frac{1}{5} \log_{2} (x^{2}) = 4$

Step 1

Multiply both sides of the equation by $5$ .

$5 (\frac{1}{5} \cdot \log_{2} (x^{2})) = 5 \cdot 4$

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 $5 (\frac{1}{5} \cdot \log_{2} (x^{2}))$ .

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Step 2.1.1.1

Combine $\frac{1}{5}$ and $\log_{2} (x^{2})$ .

$5 \frac{\log_{2} (x^{2})}{5} = 5 \cdot 4$

Step 2.1.1.2

Cancel the common factor of $5$ .

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Step 2.1.1.2.1

Cancel the common factor.

$5 \frac{\log_{2} (x^{2})}{5} = 5 \cdot 4$

Step 2.1.1.2.2

Rewrite the expression.

$\log_{2} (x^{2}) = 5 \cdot 4$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $5$ by $4$ .

$\log_{2} (x^{2}) = 20$

Step 3

Write in exponential form.

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Step 3.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 2$ , $x = x^{2}$ , and $y = 20$ .

$b = 2$

$x = x^{2}$

$y = 20$

Step 3.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$2^{20} = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{2} = 2^{20}$ .

$x^{2} = 2^{20}$

Step 4.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{2^{20}}$

Step 4.3

Simplify $\pm \sqrt{2^{20}}$ .

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Step 4.3.1

Raise $2$ to the power of $20$ .

$x = \pm \sqrt{1048576}$

Step 4.3.2

Rewrite $1048576$ as $1024^{2}$ .

$x = \pm \sqrt{1024^{2}}$

Step 4.3.3

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 1024$

Step 4.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1024$

Step 4.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1024$

Step 4.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1024, - 1024$