Trigonometry Examples

$2 \log (x) = \log (8)$

Step 1

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$\log (x^{2}) = \log (8)$

Step 2

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

$x^{2} = 8$

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.

$x = \pm \sqrt{8}$

Step 3.2

Simplify $\pm \sqrt{8}$ .

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Step 3.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 3.2.1.1

Factor $4$ out of $8$ .

$x = \pm \sqrt{4 (2)}$

Step 3.2.1.2

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

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

Step 3.2.2

Pull terms out from under the radical.

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

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.

$x = 2 \sqrt{2}$

Step 3.3.2

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

$x = - 2 \sqrt{2}$

Step 3.3.3

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

$x = 2 \sqrt{2}, - 2 \sqrt{2}$

Step 4

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

$x = 2 \sqrt{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = 2 \sqrt{2}$

Decimal Form:

$x = 2.82842712 \dots$