Precalculus Examples

$\log_{x} (9) = - 1$

Step 1

Rewrite $\log_{x} (9) = - 1$ 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 $b^{y} = x$ .

$x^{- 1} = 9$

Step 2

Solve for $x$ .

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Step 2.1

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

$\frac{1}{x} = 9$

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.

$x, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$x$

Step 2.3

Multiply each term in $\frac{1}{x} = 9$ by $x$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{1}{x} = 9$ by $x$ .

$\frac{1}{x} x = 9 x$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{1}{x} x = 9 x$

Step 2.3.2.1.2

Rewrite the expression.

$1 = 9 x$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $9 x = 1$ .

$9 x = 1$

Step 2.4.2

Divide each term in $9 x = 1$ by $9$ and simplify.

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Step 2.4.2.1

Divide each term in $9 x = 1$ by $9$ .

$\frac{9 x}{9} = \frac{1}{9}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $9$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{1}{9}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{9}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{9}$

Decimal Form:

$x = 0.\overline{1}$