Precalculus Examples

$\log_{x} (121) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$\log_{x} (121) - 2 = 0$

Step 2

Add $2$ to both sides of the equation.

$\log_{x} (121) = 2$

Step 3

Rewrite $\log_{x} (121) = 2$ 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 $b^{y} = x$ .

$x^{2} = 121$

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.

$x = \pm \sqrt{121}$

Step 4.2

Simplify $\pm \sqrt{121}$ .

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Step 4.2.1

Rewrite $121$ as $11^{2}$ .

$x = \pm \sqrt{11^{2}}$

Step 4.2.2

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

$x = \pm 11$

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.

$x = 11$

Step 4.3.2

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

$x = - 11$

Step 4.3.3

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

$x = 11, - 11$

Step 5

Exclude the solutions that do not make $\log_{x} (121) - 2 = 0$ true.

$x = 11$