Trigonometry Examples

$\log (x^{2} + 19) = 2$

Step 1

Rewrite $\log (x^{2} + 19) = 2$ 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 $b^{y} = x$ .

$10^{2} = x^{2} + 19$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x^{2} + 19 = 10^{2}$ .

$x^{2} + 19 = 10^{2}$

Step 2.2

Raise $10$ to the power of $2$ .

$x^{2} + 19 = 100$

Step 2.3

Move all terms not containing $x$ to the right side of the equation.

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Step 2.3.1

Subtract $19$ from both sides of the equation.

$x^{2} = 100 - 19$

Step 2.3.2

Subtract $19$ from $100$ .

$x^{2} = 81$

Step 2.4

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

$x = \pm \sqrt{81}$

Step 2.5

Simplify $\pm \sqrt{81}$ .

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Step 2.5.1

Rewrite $81$ as $9^{2}$ .

$x = \pm \sqrt{9^{2}}$

Step 2.5.2

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

$x = \pm 9$

Step 2.6

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

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Step 2.6.1

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

$x = 9$

Step 2.6.2

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

$x = - 9$

Step 2.6.3

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

$x = 9, - 9$