Trigonometry Examples

$f (x) = \log_{6} (2 x - 4)$

Step 1

Write $f (x) = \log_{6} (2 x - 4)$ as an equation.

$y = \log_{6} (2 x - 4)$

Step 2

Interchange the variables.

$x = \log_{6} (2 y - 4)$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $\log_{6} (2 y - 4) = x$ .

$\log_{6} (2 y - 4) = x$

Step 3.2

Rewrite $\log_{6} (2 y - 4) = x$ 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 $b^{y} = x$ .

$6^{x} = 2 y - 4$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Rewrite the equation as $2 y - 4 = 6^{x}$ .

$2 y - 4 = 6^{x}$

Step 3.3.2

Add $4$ to both sides of the equation.

$2 y = 6^{x} + 4$

Step 3.3.3

Divide each term in $2 y = 6^{x} + 4$ by $2$ and simplify.

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Step 3.3.3.1

Divide each term in $2 y = 6^{x} + 4$ by $2$ .

$\frac{2 y}{2} = \frac{6^{x}}{2} + \frac{4}{2}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $2$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{6^{x}}{2} + \frac{4}{2}$

Step 3.3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6^{x}}{2} + \frac{4}{2}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $4$ by $2$ .

$y = \frac{6^{x}}{2} + 2$

Step 4

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

$f^{- 1} (x) = \frac{6^{x}}{2} + 2$

Step 5

Verify if $f^{- 1} (x) = \frac{6^{x}}{2} + 2$ is the inverse of $f (x) = \log_{6} (2 x - 4)$ .

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Step 5.1

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

Step 5.2

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

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Step 5.2.1

Set up the composite result function.

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

Step 5.2.2

Evaluate $f^{- 1} (\log_{6} (2 x - 4))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{6^{\log_{6} (2 x - 4)}}{2} + 2$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

Exponentiation and log are inverse functions.

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{2 x - 4}{2} + 2$

Step 5.2.3.1.2

Factor $2$ out of $2 x - 4$ .

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Step 5.2.3.1.2.1

Factor $2$ out of $2 x$ .

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{2 (x) - 4}{2} + 2$

Step 5.2.3.1.2.2

Factor $2$ out of $- 4$ .

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{2 x + 2 \cdot - 2}{2} + 2$

Step 5.2.3.1.2.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{2 (x - 2)}{2} + 2$

Step 5.2.3.2

Cancel the common factor of $2$ .

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Step 5.2.3.2.1

Cancel the common factor.

$f^{- 1} (\log_{6} (2 x - 4)) = \frac{2 (x - 2)}{2} + 2$

Step 5.2.3.2.2

Divide $x - 2$ by $1$ .

$f^{- 1} (\log_{6} (2 x - 4)) = x - 2 + 2$

Step 5.2.4

Combine the opposite terms in $x - 2 + 2$ .

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Step 5.2.4.1

Add $- 2$ and $2$ .

$f^{- 1} (\log_{6} (2 x - 4)) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (\log_{6} (2 x - 4)) = x$

Step 5.3

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

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Step 5.3.1

Set up the composite result function.

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

Step 5.3.2

Evaluate $f (\frac{6^{x}}{2} + 2)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{6^{x}}{2} + 2) = \log_{6} (2 (\frac{6^{x}}{2} + 2) - 4)$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Apply the distributive property.

$f (\frac{6^{x}}{2} + 2) = \log_{6} (2 (\frac{6^{x}}{2}) + 2 \cdot 2 - 4)$

Step 5.3.3.2

Cancel the common factor of $2$ .

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Step 5.3.3.2.1

Cancel the common factor.

$f (\frac{6^{x}}{2} + 2) = \log_{6} (2 (\frac{6^{x}}{2}) + 2 \cdot 2 - 4)$

Step 5.3.3.2.2

Rewrite the expression.

$f (\frac{6^{x}}{2} + 2) = \log_{6} (6^{x} + 2 \cdot 2 - 4)$

Step 5.3.3.3

Multiply $2$ by $2$ .

$f (\frac{6^{x}}{2} + 2) = \log_{6} (6^{x} + 4 - 4)$

Step 5.3.4

Combine the opposite terms in $6^{x} + 4 - 4$ .

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Step 5.3.4.1

Subtract $4$ from $4$ .

$f (\frac{6^{x}}{2} + 2) = \log_{6} (6^{x} + 0)$

Step 5.3.4.2

Add $6^{x}$ and $0$ .

$f (\frac{6^{x}}{2} + 2) = \log_{6} (6^{x})$

Step 5.3.5

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

$f (\frac{6^{x}}{2} + 2) = x \log_{6} (6)$

Step 5.3.6

Logarithm base $6$ of $6$ is $1$ .

$f (\frac{6^{x}}{2} + 2) = x \cdot 1$

Step 5.3.7

Multiply $x$ by $1$ .

$f (\frac{6^{x}}{2} + 2) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{6^{x}}{2} + 2$ is the inverse of $f (x) = \log_{6} (2 x - 4)$ .

$f^{- 1} (x) = \frac{6^{x}}{2} + 2$