Trigonometry Examples

$45^{y}$

Step 1

Rewrite the equation as $45^{y} = x$ .

$45^{y} = x$

Step 2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (45^{y}) = \ln (x)$

Step 3

Expand $\ln (45^{y})$ by moving $y$ outside the logarithm.

$y \ln (45) = \ln (x)$

Step 4

Divide each term in $y \ln (45) = \ln (x)$ by $\ln (45)$ and simplify.

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Step 4.1

Divide each term in $y \ln (45) = \ln (x)$ by $\ln (45)$ .

$\frac{y \ln (45)}{\ln (45)} = \frac{\ln (x)}{\ln (45)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\ln (45)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{y \ln (45)}{\ln (45)} = \frac{\ln (x)}{\ln (45)}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{\ln (45)}$

Step 5

Interchange the variables.

$x = \frac{\ln (y)}{\ln (45)}$

Step 6

Solve for $y$ .

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Step 6.1

Rewrite the equation as $\frac{\ln (y)}{\ln (45)} = x$ .

$\frac{\ln (y)}{\ln (45)} = x$

Step 6.2

Multiply both sides of the equation by $\ln (45)$ .

$\ln (45) \frac{\ln (y)}{\ln (45)} = \ln (45) x$

Step 6.3

Simplify both sides of the equation.

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Step 6.3.1

Simplify the left side.

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Step 6.3.1.1

Cancel the common factor of $\ln (45)$ .

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Step 6.3.1.1.1

Cancel the common factor.

$\ln (45) \frac{\ln (y)}{\ln (45)} = \ln (45) x$

Step 6.3.1.1.2

Rewrite the expression.

$\ln (y) = \ln (45) x$

Step 6.3.2

Simplify the right side.

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Step 6.3.2.1

Reorder factors in $\ln (45) x$ .

$\ln (y) = x \ln (45)$

Step 6.4

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y)} = e^{x \ln (45)}$

Step 6.5

Rewrite $\ln (y) = x \ln (45)$ 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 $b^{y} = x$ .

$e^{x \ln (45)} = y$

Step 6.6

Rewrite the equation as $y = e^{x \ln (45)}$ .

$y = e^{x \ln (45)}$

Step 7

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

$f^{- 1} (x) = e^{x \ln (45)}$

Step 8

Verify if $f^{- 1} (x) = e^{x \ln (45)}$ is the inverse of $f (x) = \frac{\ln (x)}{\ln (45)}$ .

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Step 8.1

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

Step 8.2

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

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Step 8.2.1

Set up the composite result function.

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

Step 8.2.2

Evaluate $f^{- 1} (\frac{\ln (x)}{\ln (45)})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\frac{\ln (x)}{\ln (45)}) = e^{(\frac{\ln (x)}{\ln (45)}) \ln (45)}$

Step 8.2.3

Cancel the common factor of $\ln (45)$ .

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Step 8.2.3.1

Cancel the common factor.

$f^{- 1} (\frac{\ln (x)}{\ln (45)}) = e^{\frac{\ln (x)}{\ln (45)} \cdot \ln (45)}$

Step 8.2.3.2

Rewrite the expression.

$f^{- 1} (\frac{\ln (x)}{\ln (45)}) = e^{\ln (x)}$

Step 8.2.4

Exponentiation and log are inverse functions.

$f^{- 1} (\frac{\ln (x)}{\ln (45)}) = x$

Step 8.3

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

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Step 8.3.1

Set up the composite result function.

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

Step 8.3.2

Evaluate $f (e^{x \ln (45)})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (e^{x \ln (45)}) = \frac{\ln (e^{x \ln (45)})}{\ln (45)}$

Step 8.3.3

Expand $\ln (e^{x \ln (45)})$ by moving $x \ln (45)$ outside the logarithm.

$f (e^{x \ln (45)}) = \frac{x \ln (45) \ln (e)}{\ln (45)}$

Step 8.3.4

Cancel the common factor of $\ln (45)$ .

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Step 8.3.4.1

Cancel the common factor.

$f (e^{x \ln (45)}) = \frac{x \ln (45) \ln (e)}{\ln (45)}$

Step 8.3.4.2

Divide $x \ln (e)$ by $1$ .

$f (e^{x \ln (45)}) = x \ln (e)$

Step 8.3.5

The natural logarithm of $e$ is $1$ .

$f (e^{x \ln (45)}) = x \cdot 1$

Step 8.3.6

Multiply $x$ by $1$ .

$f (e^{x \ln (45)}) = x$

Step 8.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = e^{x \ln (45)}$ is the inverse of $f (x) = \frac{\ln (x)}{\ln (45)}$ .

$f^{- 1} (x) = e^{x \ln (45)}$