Precalculus Examples

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

Step 1

Rewrite $\log_{4} (x^{2} + 5) = 2$ 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 $b^{y} = x$ .

$4^{2} = x^{2} + 5$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x^{2} + 5 = 4^{2}$ .

$x^{2} + 5 = 4^{2}$

Step 2.2

Raise $4$ to the power of $2$ .

$x^{2} + 5 = 16$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Subtract $5$ from both sides of the equation.

$x^{2} = 16 - 5$

Step 2.3.2

Subtract $5$ from $16$ .

$x^{2} = 11$

Step 2.4

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

$x = \pm \sqrt{11}$

Step 2.5

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

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Step 2.5.1

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

$x = \sqrt{11}$

Step 2.5.2

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

$x = - \sqrt{11}$

Step 2.5.3

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

$x = \sqrt{11}, - \sqrt{11}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{11}, - \sqrt{11}$

Decimal Form:

$x = 3.31662479 \dots, - 3.31662479 \dots$