Precalculus Examples

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

Step 1

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

$y^{- 4} = 81$

Step 2

Solve for $y$ .

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Step 2.1

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

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

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

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

$y^{4}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$y^{4}$

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

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

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Step 2.3.2.1.1

Cancel the common factor.

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

Step 2.3.2.1.2

Rewrite the expression.

$1 = 81 y^{4}$

Step 2.4

Solve the equation.

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Step 2.4.1

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

$81 y^{4} = 1$

Step 2.4.2

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

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Step 2.4.2.1

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

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

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $81$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

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

Step 2.4.2.2.1.2

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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Step 2.4.4.1

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

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

Step 2.4.4.2

Any root of $1$ is $1$ .

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

Step 2.4.4.3

Simplify the denominator.

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Step 2.4.4.3.1

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

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

Step 2.4.4.3.2

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

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

Step 2.4.5

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

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Step 2.4.5.1

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

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

Step 2.4.5.2

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

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

Step 2.4.5.3

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

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

Step 3

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

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

Step 4

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$y = 0.\overline{3}$