Precalculus Examples

$4 \log_{8} (x) = 2 \log_{8} (9)$

Step 1

Simplify.

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Step 1.1

Simplify $4 \log_{8} (x)$ by moving $4$ inside the logarithm.

$\log_{8} (x^{4}) = 2 \log_{8} (9)$

Step 1.2

Simplify $2 \log_{8} (9)$ by moving $2$ inside the logarithm.

$\log_{8} (x^{4}) = \log_{8} (9^{2})$

Step 2

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

$x^{4} = 9^{2}$

Step 3

Solve for $x$ .

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Step 3.1

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

$x = \pm \sqrt[4]{9^{2}}$

Step 3.2

Simplify $\pm \sqrt[4]{9^{2}}$ .

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Step 3.2.1

Raise $9$ to the power of $2$ .

$x = \pm \sqrt[4]{81}$

Step 3.2.2

Rewrite $81$ as $3^{4}$ .

$x = \pm \sqrt[4]{3^{4}}$

Step 3.2.3

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

$x = \pm 3$

Step 3.3

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

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Step 3.3.1

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

$x = 3$

Step 3.3.2

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

$x = - 3$

Step 3.3.3

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

$x = 3, - 3$

Step 4

Exclude the solutions that do not make $4 \log_{8} (x) = 2 \log_{8} (9)$ true.

$x = 3$