Algebra Examples

$\frac{36^{x}}{6^{x}}$

Step 1

Interchange the variables.

$x = \frac{36^{y}}{6^{y}}$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\frac{36^{y}}{6^{y}} = x$ .

$\frac{36^{y}}{6^{y}} = x$

Step 2.2

Multiply both sides by $6^{y}$ .

$\frac{36^{y}}{6^{y}} \cdot 6^{y} = x \cdot 6^{y}$

Step 2.3

Simplify the left side.

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Step 2.3.1

Cancel the common factor of $6^{y}$ .

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Step 2.3.1.1

Cancel the common factor.

$\frac{36^{y}}{6^{y}} \cdot 6^{y} = x \cdot 6^{y}$

Step 2.3.1.2

Rewrite the expression.

$36^{y} = x \cdot 6^{y}$

Step 2.4

Solve for $y$ .

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Step 2.4.1

Take the log of both sides of the equation.

$\ln (36^{y}) = \ln (x \cdot 6^{y})$

Step 2.4.2

Expand $\ln (36^{y})$ by moving $y$ outside the logarithm.

$y \ln (36) = \ln (x \cdot 6^{y})$

Step 2.4.3

Rewrite $\ln (x \cdot 6^{y})$ as $\ln (x) + \ln (6^{y})$ .

$y \ln (36) = \ln (x) + \ln (6^{y})$

Step 2.4.4

Expand $\ln (6^{y})$ by moving $y$ outside the logarithm.

$y \ln (36) = \ln (x) + y \ln (6)$

Step 2.4.5

Solve the equation for $y$ .

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Step 2.4.5.1

Move all the terms containing a logarithm to the left side of the equation.

$y \ln (36) - \ln (x) - y \ln (6) = 0$

Step 2.4.5.2

Add $\ln (x)$ to both sides of the equation.

$y \ln (36) - y \ln (6) = \ln (x)$

Step 2.4.5.3

Factor $y$ out of $y \ln (36) - y \ln (6)$ .

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Step 2.4.5.3.1

Factor $y$ out of $y \ln (36)$ .

$y (\ln (36)) - y \ln (6) = \ln (x)$

Step 2.4.5.3.2

Factor $y$ out of $- y \ln (6)$ .

$y (\ln (36)) + y (- 1 \ln (6)) = \ln (x)$

Step 2.4.5.3.3

Factor $y$ out of $y (\ln (36)) + y (- 1 \ln (6))$ .

$y (\ln (36) - 1 \ln (6)) = \ln (x)$

Step 2.4.5.4

Rewrite $- 1 \ln (6)$ as $- \ln (6)$ .

$y (\ln (36) - \ln (6)) = \ln (x)$

Step 2.4.5.5

Divide each term in $y (\ln (36) - \ln (6)) = \ln (x)$ by $\ln (36) - \ln (6)$ and simplify.

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Step 2.4.5.5.1

Divide each term in $y (\ln (36) - \ln (6)) = \ln (x)$ by $\ln (36) - \ln (6)$ .

$\frac{y (\ln (36) - \ln (6))}{\ln (36) - \ln (6)} = \frac{\ln (x)}{\ln (36) - \ln (6)}$

Step 2.4.5.5.2

Simplify the left side.

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Step 2.4.5.5.2.1

Cancel the common factor of $\ln (36) - \ln (6)$ .

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Step 2.4.5.5.2.1.1

Cancel the common factor.

$\frac{y (\ln (36) - \ln (6))}{\ln (36) - \ln (6)} = \frac{\ln (x)}{\ln (36) - \ln (6)}$

Step 2.4.5.5.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{\ln (36) - \ln (6)}$

Step 3

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

$f^{- 1} (x) = \frac{\ln (x)}{\ln (36) - \ln (6)}$

Step 4

Verify if $f^{- 1} (x) = \frac{\ln (x)}{\ln (36) - \ln (6)}$ is the inverse of $f (x) = \frac{36^{x}}{6^{x}}$ .

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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 $f^{- 1} (\frac{36^{x}}{6^{x}})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{\ln (\frac{36^{x}}{6^{x}})}{\ln (36) - \ln (6)}$

Step 4.2.3

Simplify the numerator.

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Step 4.2.3.1

Use the power of quotient rule $\frac{a^{m}}{b^{m}} = {(\frac{a}{b})}^{m}$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{\ln ({(\frac{36}{6})}^{x})}{\ln (36) - \ln (6)}$

Step 4.2.3.2

Divide $36$ by $6$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{\ln (6^{x})}{\ln (36) - \ln (6)}$

Step 4.2.4

Simplify the denominator.

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Step 4.2.4.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{\ln (6^{x})}{\ln (\frac{36}{6})}$

Step 4.2.4.2

Divide $36$ by $6$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{\ln (6^{x})}{\ln (6)}$

Step 4.2.5

Expand $\ln (6^{x})$ by moving $x$ outside the logarithm.

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{x \ln (6)}{\ln (6)}$

Step 4.2.6

Cancel the common factor of $\ln (6)$ .

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Step 4.2.6.1

Cancel the common factor.

$f^{- 1} (\frac{36^{x}}{6^{x}}) = \frac{x \ln (6)}{\ln (6)}$

Step 4.2.6.2

Divide $x$ by $1$ .

$f^{- 1} (\frac{36^{x}}{6^{x}}) = x$

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 $f (\frac{\ln (x)}{\ln (36) - \ln (6)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (36) - \ln (6)}}}{6^{\frac{\ln (x)}{\ln (36) - \ln (6)}}}$

Step 4.3.3

Simplify the denominator.

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Step 4.3.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (\frac{36}{6})}}}{6^{\frac{\ln (x)}{\ln (36) - \ln (6)}}}$

Step 4.3.3.2

Divide $36$ by $6$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (6)}}}{6^{\frac{\ln (x)}{\ln (36) - \ln (6)}}}$

Step 4.3.4

Simplify the denominator.

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Step 4.3.4.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (6)}}}{6^{\frac{\ln (x)}{\ln (\frac{36}{6})}}}$

Step 4.3.4.2

Divide $36$ by $6$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (6)}}}{6^{\frac{\ln (x)}{\ln (6)}}}$

Step 4.3.5

Simplify the denominator.

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Step 4.3.5.1

Use the change of base rule $\frac{\log_{b} (x)}{\log_{b} (a)} = \log_{a} (x)$ .

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (6)}}}{6^{\log_{6} (x)}}$

Step 4.3.5.2

Exponentiation and log are inverse functions.

$f (\frac{\ln (x)}{\ln (36) - \ln (6)}) = \frac{36^{\frac{\ln (x)}{\ln (6)}}}{x}$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (x)}{\ln (36) - \ln (6)}$ is the inverse of $f (x) = \frac{36^{x}}{6^{x}}$ .

$f^{- 1} (x) = \frac{\ln (x)}{\ln (36) - \ln (6)}$