Precalculus Examples

$2 \log_{4} (x) = 3 \log_{4} (2) + \log_{4} (8)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $2 \log_{4} (x)$ by moving $2$ inside the logarithm.

$\log_{4} (x^{2}) = 3 \log_{4} (2) + \log_{4} (8)$

Step 2

Simplify the right side.

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Step 2.1

Simplify $3 \log_{4} (2) + \log_{4} (8)$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Logarithm base $4$ of $2$ is $\frac{1}{2}$ .

$\log_{4} (x^{2}) = 3 (\frac{1}{2}) + \log_{4} (8)$

Step 2.1.1.2

Combine $3$ and $\frac{1}{2}$ .

$\log_{4} (x^{2}) = \frac{3}{2} + \log_{4} (8)$

Step 2.1.1.3

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\log_{4} (x^{2}) = \frac{3}{2} + \frac{3}{2}$

Step 2.1.2

Combine fractions.

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Step 2.1.2.1

Combine the numerators over the common denominator.

$\log_{4} (x^{2}) = \frac{3 + 3}{2}$

Step 2.1.2.2

Simplify the expression.

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Step 2.1.2.2.1

Add $3$ and $3$ .

$\log_{4} (x^{2}) = \frac{6}{2}$

Step 2.1.2.2.2

Divide $6$ by $2$ .

$\log_{4} (x^{2}) = 3$

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, $b = 4$ , $x = x^{2}$ , and $y = 3$ .

$b = 4$

$x = x^{2}$

$y = 3$

Step 3.2

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

$4^{3} = x^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Rewrite the equation as $x^{2} = 4^{3}$ .

$x^{2} = 4^{3}$

Step 4.2

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

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

Step 4.3

Simplify $\pm \sqrt{4^{3}}$ .

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Step 4.3.1

Raise $4$ to the power of $3$ .

$x = \pm \sqrt{64}$

Step 4.3.2

Rewrite $64$ as $8^{2}$ .

$x = \pm \sqrt{8^{2}}$

Step 4.3.3

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

$x = \pm 8$

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.

$x = 8$

Step 4.4.2

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

$x = - 8$

Step 4.4.3

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

$x = 8, - 8$

Step 5

Exclude the solutions that do not make $2 \log_{4} (x) = 3 \log_{4} (2) + \log_{4} (8)$ true.

$x = 8$