Calculus Examples

$\ln (x^{2}) = 9$

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, $b = e$ , $x = x^{2}$ , and $y = 9$ .

$b = e$

$x = x^{2}$

$y = 9$

Step 1.2

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

$e^{9} = x^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $x^{2} = e^{9}$ .

$x^{2} = e^{9}$

Step 2.2

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

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

Step 2.3

Simplify $\pm \sqrt{e^{9}}$ .

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Step 2.3.1

Rewrite $e^{9}$ as ${(e^{4})}^{2} e$ .

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Step 2.3.1.1

Factor out $e^{8}$ .

$x = \pm \sqrt{e^{8} e}$

Step 2.3.1.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

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

Step 2.3.2

Pull terms out from under the radical.

$x = \pm e^{4} \sqrt{e}$

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.

$x = e^{4} \sqrt{e}$

Step 2.4.2

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

$x = - e^{4} \sqrt{e}$

Step 2.4.3

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

$x = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Decimal Form:

$x = 90.01713130 \dots, - 90.01713130 \dots$