Precalculus Examples

$f (x) = 6 \log (x) - 3$

Step 1

Write $f (x) = 6 \log (x) - 3$ as an equation.

$y = 6 \log (x) - 3$

Step 2

Interchange the variables.

$x = 6 \log (y) - 3$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $6 \log (y) - 3 = x$ .

$6 \log (y) - 3 = x$

Step 3.2

Add $3$ to both sides of the equation.

$6 \log (y) = x + 3$

Step 3.3

Divide each term in $6 \log (y) = x + 3$ by $6$ and simplify.

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Step 3.3.1

Divide each term in $6 \log (y) = x + 3$ by $6$ .

$\frac{6 \log (y)}{6} = \frac{x}{6} + \frac{3}{6}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $6$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{6 \log (y)}{6} = \frac{x}{6} + \frac{3}{6}$

Step 3.3.2.1.2

Divide $\log (y)$ by $1$ .

$\log (y) = \frac{x}{6} + \frac{3}{6}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Cancel the common factor of $3$ and $6$ .

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Step 3.3.3.1.1

Factor $3$ out of $3$ .

$\log (y) = \frac{x}{6} + \frac{3 (1)}{6}$

Step 3.3.3.1.2

Cancel the common factors.

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Step 3.3.3.1.2.1

Factor $3$ out of $6$ .

$\log (y) = \frac{x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 3.3.3.1.2.2

Cancel the common factor.

$\log (y) = \frac{x}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 3.3.3.1.2.3

Rewrite the expression.

$\log (y) = \frac{x}{6} + \frac{1}{2}$

Step 3.4

Rewrite $\log (y) = \frac{x}{6} + \frac{1}{2}$ 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 $b^{y} = x$ .

$10^{\frac{x}{6} + \frac{1}{2}} = y$

Step 3.5

Rewrite the equation as $y = 10^{\frac{x}{6} + \frac{1}{2}}$ .

$y = 10^{\frac{x}{6} + \frac{1}{2}}$

Step 4

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

$f^{- 1} (x) = 10^{\frac{x}{6} + \frac{1}{2}}$

Step 5

Verify if $f^{- 1} (x) = 10^{\frac{x}{6} + \frac{1}{2}}$ is the inverse of $f (x) = 6 \log (x) - 3$ .

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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 $f^{- 1} (6 \log (x) - 3)$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{6 \log (x) - 3}{6} + \frac{1}{2}}$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Cancel the common factor of $6 \log (x) - 3$ and $6$ .

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Step 5.2.3.1.1

Factor $3$ out of $6 \log (x)$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{3 (2 \log (x)) - 3}{6} + \frac{1}{2}}$

Step 5.2.3.1.2

Factor $3$ out of $- 3$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{3 (2 \log (x)) + 3 (- 1)}{6} + \frac{1}{2}}$

Step 5.2.3.1.3

Factor $3$ out of $3 (2 \log (x)) + 3 (- 1)$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{3 (2 \log (x) - 1)}{6} + \frac{1}{2}}$

Step 5.2.3.1.4

Cancel the common factors.

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Step 5.2.3.1.4.1

Factor $3$ out of $6$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{3 (2 \log (x) - 1)}{3 (2)} + \frac{1}{2}}$

Step 5.2.3.1.4.2

Cancel the common factor.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{3 (2 \log (x) - 1)}{3 \cdot 2} + \frac{1}{2}}$

Step 5.2.3.1.4.3

Rewrite the expression.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{2 \log (x) - 1}{2} + \frac{1}{2}}$

Step 5.2.3.2

Simplify $2 \log (x)$ by moving $2$ inside the logarithm.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{\log (x^{2}) - 1}{2} + \frac{1}{2}}$

Step 5.2.4

Simplify terms.

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Step 5.2.4.1

Combine the numerators over the common denominator.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{\log (x^{2}) - 1 + 1}{2}}$

Step 5.2.4.2

Combine the opposite terms in $\log (x^{2}) - 1 + 1$ .

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Step 5.2.4.2.1

Add $- 1$ and $1$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{\log (x^{2}) + 0}{2}}$

Step 5.2.4.2.2

Add $\log (x^{2})$ and $0$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{\log (x^{2})}{2}}$

Step 5.2.5

Expand $\log (x^{2})$ by moving $2$ outside the logarithm.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{2 \log (x)}{2}}$

Step 5.2.6

Cancel the common factor of $2$ .

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Step 5.2.6.1

Cancel the common factor.

$f^{- 1} (6 \log (x) - 3) = 10^{\frac{2 \log (x)}{2}}$

Step 5.2.6.2

Divide $\log (x)$ by $1$ .

$f^{- 1} (6 \log (x) - 3) = 10^{\log (x)}$

Step 5.2.7

Exponentiation and log are inverse functions.

$f^{- 1} (6 \log (x) - 3) = x$

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 $f (10^{\frac{x}{6} + \frac{1}{2}})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 \log (10^{\frac{x}{6} + \frac{1}{2}}) - 3$

Step 5.3.3

Simplify each term.

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Step 5.3.3.1

Use logarithm rules to move $\frac{x}{6} + \frac{1}{2}$ out of the exponent.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 ((\frac{x}{6} + \frac{1}{2}) \log (10)) - 3$

Step 5.3.3.2

Logarithm base $10$ of $10$ is $1$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 ((\frac{x}{6} + \frac{1}{2}) \cdot 1) - 3$

Step 5.3.3.3

Multiply $\frac{x}{6} + \frac{1}{2}$ by $1$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 (\frac{x}{6} + \frac{1}{2}) - 3$

Step 5.3.3.4

Apply the distributive property.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 (\frac{x}{6}) + 6 (\frac{1}{2}) - 3$

Step 5.3.3.5

Cancel the common factor of $6$ .

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Step 5.3.3.5.1

Cancel the common factor.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = 6 (\frac{x}{6}) + 6 (\frac{1}{2}) - 3$

Step 5.3.3.5.2

Rewrite the expression.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x + 6 (\frac{1}{2}) - 3$

Step 5.3.3.6

Cancel the common factor of $2$ .

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Step 5.3.3.6.1

Factor $2$ out of $6$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x + 2 (3) (\frac{1}{2}) - 3$

Step 5.3.3.6.2

Cancel the common factor.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x + 2 \cdot (3 (\frac{1}{2})) - 3$

Step 5.3.3.6.3

Rewrite the expression.

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x + 3 - 3$

Step 5.3.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.3.4.1

Subtract $3$ from $3$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x + 0$

Step 5.3.4.2

Add $x$ and $0$ .

$f (10^{\frac{x}{6} + \frac{1}{2}}) = x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 10^{\frac{x}{6} + \frac{1}{2}}$ is the inverse of $f (x) = 6 \log (x) - 3$ .

$f^{- 1} (x) = 10^{\frac{x}{6} + \frac{1}{2}}$