Trigonometry Examples

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

Step 1

Rewrite $\log (2 x^{2}) = 8$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{8} = 2 x^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $2 x^{2} = 10^{8}$ .

$2 x^{2} = 10^{8}$

Step 2.2

Divide each term in $2 x^{2} = 10^{8}$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x^{2} = 10^{8}$ by $2$ .

$\frac{2 x^{2}}{2} = \frac{10^{8}}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} = \frac{10^{8}}{2}$

Step 2.2.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{10^{8}}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Raise $10$ to the power of $8$ .

$x^{2} = \frac{100000000}{2}$

Step 2.2.3.2

Divide $100000000$ by $2$ .

$x^{2} = 50000000$

Step 2.3

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

$x = \pm \sqrt{50000000}$

Step 2.4

Simplify $\pm \sqrt{50000000}$ .

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Step 2.4.1

Rewrite $50000000$ as $5000^{2} \cdot 2$ .

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Step 2.4.1.1

Factor $25000000$ out of $50000000$ .

$x = \pm \sqrt{25000000 (2)}$

Step 2.4.1.2

Rewrite $25000000$ as $5000^{2}$ .

$x = \pm \sqrt{5000^{2} \cdot 2}$

Step 2.4.2

Pull terms out from under the radical.

$x = \pm 5000 \sqrt{2}$

Step 2.5

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

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Step 2.5.1

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

$x = 5000 \sqrt{2}$

Step 2.5.2

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

$x = - 5000 \sqrt{2}$

Step 2.5.3

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

$x = 5000 \sqrt{2}, - 5000 \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 5000 \sqrt{2}, - 5000 \sqrt{2}$

Decimal Form:

$x = 7071.06781186 \dots, - 7071.06781186 \dots$