Solve for x log base x of 81=-4

$\log_{x} (81) = - 4$

Step 1

Rewrite $\log_{x} (81) = - 4$ 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 $b^{y} = x$ .

$x^{- 4} = 81$

Step 2

Solve for $x$ .

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

$\frac{1}{x^{4}} = 81$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{4}, 1$

The LCM of one and any expression is the expression.

$x^{4}$

Multiply each term in $\frac{1}{x^{4}} = 81$ by $x^{4}$ to eliminate the fractions.

Multiply each term in $\frac{1}{x^{4}} = 81$ by $x^{4}$ .

$\frac{1}{x^{4}} x^{4} = 81 x^{4}$

Simplify the left side.

Cancel the common factor of $x^{4}$ .

Cancel the common factor.

$\frac{1}{x^{4}} x^{4} = 81 x^{4}$

Rewrite the expression.

$1 = 81 x^{4}$

Solve the equation.

Rewrite the equation as $81 x^{4} = 1$ .

$81 x^{4} = 1$

Divide each term in $81 x^{4} = 1$ by $81$ and simplify.

Divide each term in $81 x^{4} = 1$ by $81$ .

$\frac{81 x^{4}}{81} = \frac{1}{81}$

Simplify the left side.

Cancel the common factor of $81$ .

Cancel the common factor.

$\frac{81 x^{4}}{81} = \frac{1}{81}$

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{1}{81}$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{\frac{1}{81}}$

Simplify $\pm \sqrt[4]{\frac{1}{81}}$ .

Rewrite $\sqrt[4]{\frac{1}{81}}$ as $\frac{\sqrt[4]{1}}{\sqrt[4]{81}}$ .

$x = \pm \frac{\sqrt[4]{1}}{\sqrt[4]{81}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt[4]{81}}$

Simplify the denominator.

Rewrite $81$ as $3^{4}$ .

$x = \pm \frac{1}{\sqrt[4]{3^{4}}}$

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

$x = \pm \frac{1}{3}$

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

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

$x = \frac{1}{3}$

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

$x = - \frac{1}{3}$

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

$x = \frac{1}{3}, - \frac{1}{3}$

Step 3

Exclude the solutions that do not make $\log_{x} (81) = - 4$ true.

$x = \frac{1}{3}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{3}$

Decimal Form:

$x = 0.\overline{3}$

Solve for x log base x of 81=-4

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