Precalculus Examples

$4 \ln (x) = \ln (8) + \ln (2)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $\ln (8) + \ln (2)$ .

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Step 2.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x^{4}) = \ln (8 \cdot 2)$

Step 2.1.2

Multiply $8$ by $2$ .

$\ln (x^{4}) = \ln (16)$

Step 3

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

$x^{4} = 16$

Step 4

Solve for $x$ .

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Step 4.1

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

$x = \pm \sqrt[4]{16}$

Step 4.2

Simplify $\pm \sqrt[4]{16}$ .

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Step 4.2.1

Rewrite $16$ as $2^{4}$ .

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

Step 4.2.2

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

$x = \pm 2$

Step 4.3

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

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Step 4.3.1

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

$x = 2$

Step 4.3.2

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

$x = - 2$

Step 4.3.3

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

$x = 2, - 2$

Step 5

Exclude the solutions that do not make $4 \ln (x) = \ln (8) + \ln (2)$ true.

$x = 2$