Precalculus Examples

$\log_{5} (x^{2}) = \log_{5} (6^{3})$

Step 1

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

$x^{2} = 6^{3}$

Step 2

Solve for $x$ .

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Step 2.1

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

$x = \pm \sqrt{6^{3}}$

Step 2.2

Simplify $\pm \sqrt{6^{3}}$ .

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Step 2.2.1

Raise $6$ to the power of $3$ .

$x = \pm \sqrt{216}$

Step 2.2.2

Rewrite $216$ as $6^{2} \cdot 6$ .

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Step 2.2.2.1

Factor $36$ out of $216$ .

$x = \pm \sqrt{36 (6)}$

Step 2.2.2.2

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

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

Step 2.2.3

Pull terms out from under the radical.

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

Step 2.3

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

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Step 2.3.1

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

$x = 6 \sqrt{6}$

Step 2.3.2

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

$x = - 6 \sqrt{6}$

Step 2.3.3

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

$x = 6 \sqrt{6}, - 6 \sqrt{6}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 6 \sqrt{6}, - 6 \sqrt{6}$

Decimal Form:

$x = 14.69693845 \dots, - 14.69693845 \dots$