Calculus Examples

$e^{\ln (x^{2})} - 16 = 0$

Step 1

Add $16$ to both sides of the equation.

$e^{\ln (x^{2})} = 16$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\ln (x^{2})}) = \ln (16)$

Step 3

Expand the left side.

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Step 3.1

Expand $\ln (e^{\ln (x^{2})})$ by moving $\ln (x^{2})$ outside the logarithm.

$\ln (x^{2}) \ln (e) = \ln (16)$

Step 3.2

The natural logarithm of $e$ is $1$ .

$\ln (x^{2}) \cdot 1 = \ln (16)$

Step 3.3

Multiply $\ln (x^{2})$ by $1$ .

$\ln (x^{2}) = \ln (16)$

Step 4

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x^{2})} = e^{\ln (16)}$

Step 5

Rewrite $\ln (x^{2}) = \ln (16)$ 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 $b^{y} = x$ .

$e^{\ln (16)} = x^{2}$

Step 6

Solve for $x$ .

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Step 6.1

Rewrite the equation as $x^{2} = e^{\ln (16)}$ .

$x^{2} = e^{\ln (16)}$

Step 6.2

Exponentiation and log are inverse functions.

$x^{2} = 16$

Step 6.3

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

$x = \pm \sqrt{16}$

Step 6.4

Simplify $\pm \sqrt{16}$ .

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Step 6.4.1

Rewrite $16$ as $4^{2}$ .

$x = \pm \sqrt{4^{2}}$

Step 6.4.2

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

$x = \pm 4$

Step 6.5

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

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Step 6.5.1

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

$x = 4$

Step 6.5.2

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

$x = - 4$

Step 6.5.3

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

$x = 4, - 4$