Precalculus Examples

$\frac{(\frac{4}{x}) x^{4} - 4 x^{3} (4 \ln (x))}{x^{8}} = 0$

Step 1

Factor $x^{3}$ out of $(\frac{4}{x}) x^{4} - 4 x^{3} (4 \ln (x))$ .

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Step 1.1

Factor $x^{3}$ out of $(\frac{4}{x}) x^{4}$ .

$\frac{x^{3} (\frac{4}{x} \cdot x) - 4 x^{3} (4 \ln (x))}{x^{8}} = 0$

Step 1.2

Factor $x^{3}$ out of $- 4 x^{3} (4 \ln (x))$ .

$\frac{x^{3} (\frac{4}{x} \cdot x) + x^{3} (- 4 (4 \ln (x)))}{x^{8}} = 0$

Step 1.3

Factor $x^{3}$ out of $x^{3} (\frac{4}{x} x) + x^{3} (- 4 (4 \ln (x)))$ .

$\frac{x^{3} (\frac{4}{x} \cdot x - 4 (4 \ln (x)))}{x^{8}} = 0$

Step 2

Cancel the common factor of $x$ .

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Step 2.1

Cancel the common factor.

$\frac{x^{3} (\frac{4}{x} \cdot x - 4 (4 \ln (x)))}{x^{8}} = 0$

Step 2.2

Rewrite the expression.

$\frac{x^{3} (4 - 4 (4 \ln (x)))}{x^{8}} = 0$

Step 3

Simplify $4 \ln (x)$ by moving $4$ inside the logarithm.

$\frac{x^{3} (4 - 4 \ln (x^{4}))}{x^{8}} = 0$

Step 4

Simplify $- 4 \ln (x^{4})$ by moving $4$ inside the logarithm.

$\frac{x^{3} (4 - \ln ({(x^{4})}^{4}))}{x^{8}} = 0$

Step 5

Multiply the exponents in ${(x^{4})}^{4}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{x^{3} (4 - \ln (x^{4 \cdot 4}))}{x^{8}} = 0$

Step 5.2

Multiply $4$ by $4$ .

$\frac{x^{3} (4 - \ln (x^{16}))}{x^{8}} = 0$

Step 6

Reduce the expression $\frac{x^{3} (4 - \ln (x^{16}))}{x^{8}}$ by cancelling the common factors.

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Step 6.1

Factor $x^{3}$ out of $x^{8}$ .

$\frac{x^{3} (4 - \ln (x^{16}))}{x^{3} x^{5}} = 0$

Step 6.2

Cancel the common factor.

$\frac{x^{3} (4 - \ln (x^{16}))}{x^{3} x^{5}} = 0$

Step 6.3

Rewrite the expression.

$\frac{4 - \ln (x^{16})}{x^{5}} = 0$

Step 7

Set the numerator equal to zero.

$4 - \ln (x^{16}) = 0$

Step 8

Solve the equation for $x$ .

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Step 8.1

Subtract $4$ from both sides of the equation.

$- \ln (x^{16}) = - 4$

Step 8.2

Divide each term in $- \ln (x^{16}) = - 4$ by $- 1$ and simplify.

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Step 8.2.1

Divide each term in $- \ln (x^{16}) = - 4$ by $- 1$ .

$\frac{- \ln (x^{16})}{- 1} = \frac{- 4}{- 1}$

Step 8.2.2

Simplify the left side.

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Step 8.2.2.1

Dividing two negative values results in a positive value.

$\frac{\ln (x^{16})}{1} = \frac{- 4}{- 1}$

Step 8.2.2.2

Divide $\ln (x^{16})$ by $1$ .

$\ln (x^{16}) = \frac{- 4}{- 1}$

Step 8.2.3

Simplify the right side.

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Step 8.2.3.1

Divide $- 4$ by $- 1$ .

$\ln (x^{16}) = 4$

Step 8.3

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

$e^{\ln (x^{16})} = e^{4}$

Step 8.4

Rewrite $\ln (x^{16}) = 4$ 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 $b^{y} = x$ .

$e^{4} = x^{16}$

Step 8.5

Solve for $x$ .

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Step 8.5.1

Rewrite the equation as $x^{16} = e^{4}$ .

$x^{16} = e^{4}$

Step 8.5.2

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

$x = \pm \sqrt[16]{e^{4}}$

Step 8.5.3

Simplify $\pm \sqrt[16]{e^{4}}$ .

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Step 8.5.3.1

Rewrite $\sqrt[16]{e^{4}}$ as $\sqrt[4]{\sqrt[4]{e^{4}}}$ .

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

Step 8.5.3.2

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

$x = \pm \sqrt[4]{e}$

Step 8.5.4

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

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Step 8.5.4.1

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

$x = \sqrt[4]{e}$

Step 8.5.4.2

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

$x = - \sqrt[4]{e}$

Step 8.5.4.3

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

$x = \sqrt[4]{e}, - \sqrt[4]{e}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt[4]{e}, - \sqrt[4]{e}$

Decimal Form:

$x = 1.28402541 \dots, - 1.28402541 \dots$