Calculus Examples

$\ln ({(x - 2)}^{2}) = 14$

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, $b = e$ , $x = {(x - 2)}^{2}$ , and $y = 14$ .

$b = e$

$x = {(x - 2)}^{2}$

$y = 14$

Step 1.2

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

$e^{14} = {(x - 2)}^{2}$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as ${(x - 2)}^{2} = e^{14}$ .

${(x - 2)}^{2} = e^{14}$

Step 2.2

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

$x - 2 = \pm \sqrt{e^{14}}$

Step 2.3

Simplify $\pm \sqrt{e^{14}}$ .

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Step 2.3.1

Rewrite $e^{14}$ as ${(e^{7})}^{2}$ .

$x - 2 = \pm \sqrt{{(e^{7})}^{2}}$

Step 2.3.2

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

$x - 2 = \pm e^{7}$

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.

$x - 2 = e^{7}$

Step 2.4.2

Add $2$ to both sides of the equation.

$x = e^{7} + 2$

Step 2.4.3

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

$x - 2 = - e^{7}$

Step 2.4.4

Add $2$ to both sides of the equation.

$x = - e^{7} + 2$

Step 2.4.5

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

$x = e^{7} + 2, - e^{7} + 2$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = e^{7} + 2, - e^{7} + 2$

Decimal Form:

$x = 1098.63315842 \dots, - 1094.63315842 \dots$