Precalculus Examples

$\ln ({(x)}^{\ln (x)}) = 4$

Step 1

Expand $\ln (x^{\ln (x)})$ .

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Step 1.1

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

$\ln (x) \ln (x)$

Step 1.2

Raise $\ln (x)$ to the power of $1$ .

$\ln^{1} (x) \ln (x)$

Step 1.3

Raise $\ln (x)$ to the power of $1$ .

$\ln^{1} (x) \ln^{1} (x)$

Step 1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\ln (x)}^{1 + 1}$

Step 1.5

Add $1$ and $1$ .

$\ln^{2} (x)$

Step 2

The expanded equation is $\ln^{2} (x) = 4$ .

$\ln^{2} (x) = 4$

Step 3

Solve for $x$ .

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Step 3.1

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

$\ln (x) = \pm \sqrt{4}$

Step 3.2

Simplify $\pm \sqrt{4}$ .

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Step 3.2.1

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

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

Step 3.2.2

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

$\ln (x) = \pm 2$

Step 3.3

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

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Step 3.3.1

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

$\ln (x) = 2$

Step 3.3.2

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

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

Step 3.3.3

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

$e^{2} = x$

Step 3.3.4

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

$x = e^{2}$

Step 3.3.5

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

$\ln (x) = - 2$

Step 3.3.6

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

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

Step 3.3.7

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

$e^{- 2} = x$

Step 3.3.8

Solve for $x$ .

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Step 3.3.8.1

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

$x = e^{- 2}$

Step 3.3.8.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x = \frac{1}{e^{2}}$

Step 3.3.9

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

$x = e^{2}, \frac{1}{e^{2}}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = e^{2}, \frac{1}{e^{2}}$

Decimal Form:

$x = 7.38905609 \dots, 0.13533528 \dots$