Precalculus Examples

$\log (X^{2}) = 6$

Step 1

Write in exponential form.

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Step 1.1

For logarithmic equations, $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ such that $x > 0$ , $b > 0$ , and $b \neq 1$ . In this case, $b = 10$ , $x = X^{2}$ , and $y = 6$ .

$b = 10$

$x = X^{2}$

$y = 6$

Step 1.2

Substitute the values of $b$ , $x$ , and $y$ into the equation $b^{y} = x$ .

$10^{6} = X^{2}$

Step 2

Solve for $X$ .

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Step 2.1

Rewrite the equation as $X^{2} = 10^{6}$ .

$X^{2} = 10^{6}$

Step 2.2

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

$X = \pm \sqrt{10^{6}}$

Step 2.3

Simplify $\pm \sqrt{10^{6}}$ .

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Step 2.3.1

Raise $10$ to the power of $6$ .

$X = \pm \sqrt{1000000}$

Step 2.3.2

Rewrite $1000000$ as $1000^{2}$ .

$X = \pm \sqrt{1000^{2}}$

Step 2.3.3

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

$X = \pm 1000$

Step 2.4

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

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Step 2.4.1

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

$X = 1000$

Step 2.4.2

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

$X = - 1000$

Step 2.4.3

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

$X = 1000, - 1000$