Precalculus Examples

$\log_{6} (x^{2} + 5) = \log_{6} (41)$

Step 1

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

$x^{2} + 5 = 41$

Step 2

Solve for $x$ .

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Step 2.1

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

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Step 2.1.1

Subtract $5$ from both sides of the equation.

$x^{2} = 41 - 5$

Step 2.1.2

Subtract $5$ from $41$ .

$x^{2} = 36$

Step 2.2

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

$x = \pm \sqrt{36}$

Step 2.3

Simplify $\pm \sqrt{36}$ .

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Step 2.3.1

Rewrite $36$ as $6^{2}$ .

$x = \pm \sqrt{6^{2}}$

Step 2.3.2

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

$x = \pm 6$

Step 2.4

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

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Step 2.4.1

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

$x = 6$

Step 2.4.2

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

$x = - 6$

Step 2.4.3

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

$x = 6, - 6$