Algebra Examples

$y = \log_{12} (\frac{1}{x})$

Step 1

Interchange the variables.

$x = \log_{12} (\frac{1}{y})$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\log_{12} (\frac{1}{y}) = x$ .

$\log_{12} (\frac{1}{y}) = x$

Step 2.2

Rewrite $\log_{12} (\frac{1}{y}) = x$ 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 $b^{y} = x$ .

$12^{x} = \frac{1}{y}$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Rewrite the equation as $\frac{1}{y} = 12^{x}$ .

$\frac{1}{y} = 12^{x}$

Step 2.3.2

Find the LCD of the terms in the equation.

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Step 2.3.2.1

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

$y, 1$

Step 2.3.2.2

The LCM of one and any expression is the expression.

$y$

Step 2.3.3

Multiply each term in $\frac{1}{y} = 12^{x}$ by $y$ to eliminate the fractions.

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Step 2.3.3.1

Multiply each term in $\frac{1}{y} = 12^{x}$ by $y$ .

$\frac{1}{y} y = 12^{x} y$

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Cancel the common factor of $y$ .

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Step 2.3.3.2.1.1

Cancel the common factor.

$\frac{1}{y} y = 12^{x} y$

Step 2.3.3.2.1.2

Rewrite the expression.

$1 = 12^{x} y$

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Reorder factors in $12^{x} y$ .

$1 = y \cdot 12^{x}$

Step 2.3.4

Solve the equation.

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Step 2.3.4.1

Rewrite the equation as $y \cdot 12^{x} = 1$ .

$y \cdot 12^{x} = 1$

Step 2.3.4.2

Divide each term in $y \cdot 12^{x} = 1$ by $12^{x}$ and simplify.

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Step 2.3.4.2.1

Divide each term in $y \cdot 12^{x} = 1$ by $12^{x}$ .

$\frac{y \cdot 12^{x}}{12^{x}} = \frac{1}{12^{x}}$

Step 2.3.4.2.2

Simplify the left side.

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Step 2.3.4.2.2.1

Cancel the common factor of $12^{x}$ .

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Step 2.3.4.2.2.1.1

Cancel the common factor.

$\frac{y \cdot 12^{x}}{12^{x}} = \frac{1}{12^{x}}$

Step 2.3.4.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{1}{12^{x}}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = \frac{1}{12^{x}}$

Step 4

Verify if $f^{- 1} (x) = \frac{1}{12^{x}}$ is the inverse of $f (x) = \log_{12} (\frac{1}{x})$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\log_{12} (\frac{1}{x}))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\log_{12} (\frac{1}{x})) = \frac{1}{12^{\log_{12} (\frac{1}{x})}}$

Step 4.2.3

Exponentiation and log are inverse functions.

$f^{- 1} (\log_{12} (\frac{1}{x})) = \frac{1}{\frac{1}{x}}$

Step 4.2.4

Multiply the numerator by the reciprocal of the denominator.

$f^{- 1} (\log_{12} (\frac{1}{x})) = 1 x$

Step 4.2.5

Multiply $x$ by $1$ .

$f^{- 1} (\log_{12} (\frac{1}{x})) = x$

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (\frac{1}{12^{x}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{1}{12^{x}}) = \log_{12} (\frac{1}{\frac{1}{12^{x}}})$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$f (\frac{1}{12^{x}}) = \log_{12} (1 \cdot 12^{x})$

Step 4.3.4

Rewrite $\log_{12} (1 \cdot 12^{x})$ as $\log_{12} (1) + \log_{12} (12^{x})$ .

$f (\frac{1}{12^{x}}) = \log_{12} (1) + \log_{12} (12^{x})$

Step 4.3.5

Use logarithm rules to move $x$ out of the exponent.

$f (\frac{1}{12^{x}}) = \log_{12} (1) + x \log_{12} (12)$

Step 4.3.6

Logarithm base $12$ of $12$ is $1$ .

$f (\frac{1}{12^{x}}) = \log_{12} (1) + x \cdot 1$

Step 4.3.7

Multiply $x$ by $1$ .

$f (\frac{1}{12^{x}}) = \log_{12} (1) + x$

Step 4.3.8

Logarithm base $12$ of $1$ is $0$ .

$f (\frac{1}{12^{x}}) = 0 + x$

Step 4.3.9

Add $0$ and $x$ .

$f (\frac{1}{12^{x}}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{1}{12^{x}}$ is the inverse of $f (x) = \log_{12} (\frac{1}{x})$ .

$f^{- 1} (x) = \frac{1}{12^{x}}$