Trigonometry Examples

$y = \log_{4} (4 x)$

Step 1

Interchange the variables.

$x = \log_{4} (4 y)$

Step 2

Solve for $y$ .

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Step 2.1

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

$\log_{4} (4 y) = x$

Step 2.2

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

$4^{x} = 4 y$

Step 2.3

Solve for $y$ .

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Step 2.3.1

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

$4 y = 4^{x}$

Step 2.3.2

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

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Step 2.3.2.1

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

$\frac{4 y}{4} = \frac{4^{x}}{4}$

Step 2.3.2.2

Simplify the left side.

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Step 2.3.2.2.1

Cancel the common factor of $4$ .

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Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{4 y}{4} = \frac{4^{x}}{4}$

Step 2.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4^{x}}{4}$

Step 2.3.2.3

Simplify the right side.

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Step 2.3.2.3.1

Cancel the common factor of $4^{x}$ and $4$ .

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Step 2.3.2.3.1.1

Factor $4$ out of $4^{x}$ .

$y = \frac{4 \cdot 4^{x - 1}}{4}$

Step 2.3.2.3.1.2

Cancel the common factors.

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Step 2.3.2.3.1.2.1

Factor $4$ out of $4$ .

$y = \frac{4 \cdot 4^{x - 1}}{4 (1)}$

Step 2.3.2.3.1.2.2

Cancel the common factor.

$y = \frac{4 \cdot 4^{x - 1}}{4 \cdot 1}$

Step 2.3.2.3.1.2.3

Rewrite the expression.

$y = \frac{4^{x - 1}}{1}$

Step 2.3.2.3.1.2.4

Divide $4^{x - 1}$ by $1$ .

$y = 4^{x - 1}$

Step 3

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

$f^{- 1} (x) = 4^{x - 1}$

Step 4

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

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

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

Step 4.2.3

Remove parentheses.

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

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 $f (4^{x - 1})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Rewrite $\log_{4} (4 (4^{x - 1}))$ as $\log_{4} (4) + \log_{4} (4^{x - 1})$ .

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

Step 4.3.4

Use logarithm rules to move $x - 1$ out of the exponent.

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

Step 4.3.5

Logarithm base $4$ of $4$ is $1$ .

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

Step 4.3.6

Multiply $x - 1$ by $1$ .

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

Step 4.3.7

Logarithm base $4$ of $4$ is $1$ .

$f (4^{x - 1}) = 1 + x - 1$

Step 4.3.8

Combine the opposite terms in $1 + x - 1$ .

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Step 4.3.8.1

Subtract $1$ from $1$ .

$f (4^{x - 1}) = x + 0$

Step 4.3.8.2

Add $x$ and $0$ .

$f (4^{x - 1}) = x$

Step 4.4

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

$f^{- 1} (x) = 4^{x - 1}$