Exponential Function Manipulation - AP Precalculus

1. What is the inverse of the exponential function $f(x) = a^x$?

   $f^{-1}(x) = \text{log}_a(x)$. Exponential and logarithmic functions are inverse pairs.

2. How do you express $e^{\text{ln}(x)}$ in terms of $x$?

   $x$. The exponential and natural logarithm functions are inverses.

3. What is the expression $\text{e}^{x+y}$ equal to in exponential terms?

   $\text{e}^x \times \text{e}^y$. Using the product rule for exponentials with the same base.

4. What is the general form of an exponential function?

   $f(x) = a \times b^x$. Standard exponential form where $a$ is the initial value and $b$ is the base.

5. What is the y-intercept of $f(x) = a \times b^x$?

   $a$. When $x = 0$, $f(0) = a \times b^0 = a \times 1 = a$.

6. What is the value of $f(x) = 4 \times (0.5)^x$ at $x = 2$?

   $1$. Calculate: $4 \times (0.5)^2 = 4 \times 0.25 = 1$.

7. Evaluate $3^x = 243$ for $x$.

   $x = 5$. Since $3^5 = 243$, we have $x = 5$.

8. What property of exponents is used in $a^x \times a^y = a^{x+y}$?

   Product of powers. When multiplying powers with the same base, add the exponents.

9. Convert the expression $x = \ln(y)$ to exponential form.

   $e^x = y$. Converting from logarithmic to exponential form.

10. What is $\text{ln}(\text{e}^x)$?

    $x$. Natural logarithm and $e$ are inverse functions.

11. What is the range of the function $f(x) = 3^x$?

    $(0, \text{inf})$. Exponential functions with positive bases have all positive outputs.

12. What is the expression for $f(x) = 2 \times b^x$ when $f(x) = 16$?

    $2 \times b^x = 16$. Setting the function equal to the given value creates an equation.

13. What is the base $b$ in $f(x) = b^x$ for $b > 0$ and $b \neq 1$?

    $b$ is a positive constant not equal to 1. Base must be positive and not equal to 1 for exponential functions.

14. What is the expression for $e^{2\text{ln}(x)}$?

    $x^2$. Using the property $e^{\ln(x)} = x$ and power rule.

15. What is $\text{log}_b(b^x)$ equal to?

    $x$. The logarithm and exponential with the same base cancel out.

16. What is the exponential form of $x = \text{log}_b(y)$?

    $b^x = y$. Converting from logarithmic to exponential form.

17. What does the expression $b^{x-y}$ simplify to?

    $\frac{b^x}{b^y}$. Subtracting exponents equals division of the same base powers.

18. Which rule is used to solve $b^{x+y} = b^x \times b^y$?

    Exponential rule for addition. This is the fundamental property for combining exponential expressions.

19. What is the derivative of $f(x) = e^x$?

    $f'(x) = e^x$. The natural exponential function is its own derivative.

20. What is the simplified form of $b^0$?

    $1$. Any non-zero number raised to the power 0 equals 1.

21. How do you express $\text{log}_a(a^x)$?

    $x$. The logarithm and exponential with the same base cancel out.

22. How do you solve $e^x = 5$?

    $x = \text{ln}(5)$. Taking the natural logarithm of both sides isolates $x$.

23. How do you solve $3^x = 27$?

    $x = 3$. Since $3^3 = 27$, we have $x = 3$.

24. Rewrite the expression $b^{-x}$ in terms of fractions.

    $\frac{1}{b^x}$. Negative exponents represent reciprocals of positive exponents.

25. Convert $8 = 2^x$ into a logarithmic equation.

    $x = \text{log}_2(8)$. Converting from exponential to logarithmic form.

26. What is $\text{log}_a(1)$ equal to?

    $0$. Any number raised to the power 0 equals 1, so $\log_a(1) = 0$.

27. Evaluate $2^x = 32$ for $x$.

    $x = 5$. Since $2^5 = 32$, we have $x = 5$.

28. What is the simplified form of $(b^x)^0$?

    $1$. Any expression raised to the power 0 equals 1.

29. What is the value of $f(x) = 5 \times 2^x$ when $x = 3$?

    $40$. Substitute $x = 3$: $f(3) = 5 \times 2^3 = 5 \times 8 = 40$.

30. Find the value of $x$ in $10^x = 1000$.

    $x = 3$. Since $10^3 = 1000$, we have $x = 3$.

31. What is $f(x) = 2^x$ expressed as a logarithmic function?

    $x = \text{log}_2(f(x))$. Taking the logarithm base 2 of both sides isolates $x$.

32. What is the value of $x$ if $2^x = 1$?

    $x = 0$. Any positive number raised to power 0 equals 1.

33. How do you rewrite $4^x = 16$ using logarithms?

    $x = \text{log}_4(16)$. Converting exponential to logarithmic form by taking log of both sides.

34. What is the simplified form of $\text{log}_b(b)$?

    $1$. The logarithm of a base to itself equals 1.

35. Convert the expression $x = \text{ln}(y)$ to exponential form.

    $\text{e}^x = y$. Converting from logarithmic to exponential form.

36. What is $b^1$ equal to?

    $b$. Any number raised to the power 1 equals itself.

37. What is the domain of $f(x) = 2^x$?

    $(-\text{inf}, \text{inf})$. Exponential functions accept all real number inputs.

38. Express $b^{x+y}$ in its expanded form.

    $b^x \times b^y$. The sum rule for exponents breaks into separate factors.

39. Identify the base in the function $f(x) = 3 \times 2^x$.

    2. The base is the number being raised to the power $x$.

40. Simplify the expression $(b^x)^y$.

    $b^{xy}$. Power of a power rule: multiply the exponents.

41. What is the inverse of $f(x) = \text{e}^x$?

    $f^{-1}(x) = \text{ln}(x)$. The natural exponential and logarithm are inverse functions.

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