Inverses of Exponential Functions - AP Precalculus

1. What operation is the inverse of exponentiation?

   The inverse operation is taking the logarithm. Logarithm is the inverse operation of exponentiation.

2. Identify the inverse of $f(x) = a^x$.

   The inverse is $f^{-1}(x) = \text{log}_a(x)$. Base-$a$ logarithm undoes base-$a$ exponentiation.

3. Identify the inverse of $f(x) = 9^{x-4}$.

   The inverse is $f^{-1}(x) = \text{log}_9(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.

4. Find the inverse of $f(x) = e^{x+2}$.

   The inverse is $f^{-1}(x) = \text{ln}(x) - 2$. Addition in exponent becomes subtraction in inverse.

5. Find the inverse of $f(x) = 12^{2x+1}$.

   The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_{12}(x) - \frac{1}{2}$. Factor out coefficient from exponent terms.

6. Identify the inverse of $f(x) = b^{2x+1}$.

   The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_b(x) - \frac{1}{2}$. Factor out coefficient and shift from exponent.

7. What is the inverse of $f(x) = 10^{3x}$?

   The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_{10}(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.

8. Which function is the inverse of $f(x) = 4^{x-3}$?

   The inverse is $f^{-1}(x) = \text{log}_4(x) + 3$. Subtract 3 from exponent, so add 3 to inverse.

9. Identify the steps to find inverse of $f(x) = a^x$.

   Switch $x$ and $y$, solve for $y$. Result: $y = \text{log}_a(x)$. Standard method for finding inverse functions.

10. Identify the inverse of $f(x) = 2^x$.

    The inverse is $f^{-1}(x) = \text{log}_2(x)$. Base-2 logarithm undoes base-2 exponentiation.

11. Identify the inverse of $f(x) = 15^{x-2}$.

    The inverse is $f^{-1}(x) = \text{log}_{15}(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.

12. Find the inverse of $f(x) = 7^{x-1}$.

    The inverse is $f^{-1}(x) = \text{log}_7(x) + 1$. Subtract 1 from exponent, so add 1 to inverse.

13. What is the inverse of the natural exponential function?

    The inverse is the natural logarithm function. Natural log $\ln(x)$ undoes $e^x$.

14. Find the inverse of $f(x) = 6^{x+1}$.

    The inverse is $f^{-1}(x) = \text{log}_6(x) - 1$. Addition in exponent becomes subtraction in inverse.

15. Which is the inverse of $f(x) = 9^{x-2}$?

    The inverse is $f^{-1}(x) = \text{log}_9(x) + 2$. Subtract 2 from exponent, so add 2 to inverse.

16. Convert $y = 2^{x-5}$ to its inverse function.

    The inverse is $y = \text{log}_2(x) + 5$. Subtract 5 from exponent, so add 5 to inverse.

17. What is the inverse of the function $f(x) = e^x$?

    The inverse is $f^{-1}(x) = \text{ln}(x)$. Natural log undoes the natural exponential function.

18. Find the inverse of $f(x) = 5^x$.

    The inverse is $f^{-1}(x) = \text{log}_5(x)$. Base-5 logarithm undoes base-5 exponentiation.

19. What is the inverse of $f(x) = 2^{3x}$?

    The inverse is $f^{-1}(x) = \frac{1}{3} \text{log}_2(x)$. Coefficient 3 in exponent becomes divisor $\frac{1}{3}$.

20. State the property used to find inverses of exponential functions.

    Exponential and logarithmic functions are inverses. Each function undoes the other's operation.

21. Convert $y = e^{x-1}$ to its inverse function.

    The inverse is $y = \text{ln}(x) + 1$. Subtract 1 from exponent, so add 1 to inverse.

22. What is the inverse function of $f(x) = e^{2x}$?

    The inverse is $f^{-1}(x) = \frac{1}{2} \text{ln}(x)$. Coefficient in exponent becomes divisor in inverse.

23. What is the inverse of $f(x) = 8^{2x}$?

    The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_8(x)$. Coefficient 2 in exponent becomes divisor $\frac{1}{2}$.

24. Identify the inverse of $f(x) = 7^{3x-1}$.

    The inverse is $f^{-1}(x) = \frac{1}{3} (\text{log}_7(x) + 1)$. Factor out coefficient from both terms in exponent.

25. What is the inverse of $f(x) = 13^x$?

    The inverse is $f^{-1}(x) = \text{log}_{13}(x)$. Base-13 logarithm undoes base-13 exponentiation.

26. Convert $y = 5^{x-4}$ to its inverse function.

    The inverse is $y = \text{log}_5(x) + 4$. Subtract 4 from exponent, so add 4 to inverse.

27. Which function is the inverse of $f(x) = 3^x$?

    The inverse is $f^{-1}(x) = \text{log}_3(x)$. Base-3 logarithm undoes base-3 exponentiation.

28. What is the inverse of $f(x) = b^x$?

    The inverse is $f^{-1}(x) = \text{log}_b(x)$. General base-$b$ logarithm undoes base-$b$ exponentiation.

29. Determine the inverse of $f(x) = 2^{x+3}$.

    The inverse is $f^{-1}(x) = \text{log}_2(x) - 3$. Addition in exponent becomes subtraction in inverse.

30. Find the inverse of $f(x) = 3^{x+5}$.

    The inverse is $f^{-1}(x) = \text{log}_3(x) - 5$. Addition in exponent becomes subtraction in inverse.

31. Convert $y = 4^x$ to its inverse function.

    The inverse is $x = 4^y$ or $y = \text{log}_4(x)$. Switch variables and solve for $y$.

32. Find the inverse of $f(x) = 3^{2x}$.

    The inverse is $f^{-1}(x) = \frac{1}{2} \text{log}_3(x)$. Coefficient 2 in exponent becomes divisor $\frac{1}{2}$.

33. Find the inverse of $f(x) = 11^{x+2}$.

    The inverse is $f^{-1}(x) = \text{log}_{11}(x) - 2$. Addition in exponent becomes subtraction in inverse.

34. What is the inverse of $f(x) = 10^x$?

    The inverse is $f^{-1}(x) = \text{log}_{10}(x)$. Common logarithm undoes base-10 exponentiation.

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