Exponential and Logarithmic Equations and Inequalities - AP Precalculus

1. Find the solution: $2^x = 16$.

   $x = 4$. Since $16 = 2^4$, we have $x = 4$.

2. Solve the equation: $\text{ln}(x) = 2$.

   $x = e^2$. Convert to exponential form: $e^2 = x$.

3. Identify the change of base formula for logarithms.

   $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Converts logarithm from base $b$ to any other base $k$.

4. Simplify: $e^{\text{ln}(x)}$.

   $x$. Exponential and natural logarithm cancel as inverse functions.

5. What is the value of $\text{log}_b(b^0)$?

   0. Since $b^0 = 1$ and $\log_b(1) = 0$.

6. Solve for $x$: $e^x = 1$.

   $x = 0$. Any number to the power of 0 equals 1.

7. What is the base change formula for $\text{log}_b(x)$?

   $\text{log}_b(x) = \frac{\text{ln}(x)}{\text{ln}(b)}$. Uses natural logarithm to convert between bases.

8. Solve for $x$: $4^x = 64$.

   $x = 3$. Since $64 = 4^3$, we have $x = 3$.

9. What does the equation $b^{\log_b(x)} = x$ represent?

   Identity property of logarithms. Exponential and logarithm cancel each other as inverse functions.

10. Solve the equation: $x = \text{log}_2(32)$.

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

11. Solve for $x$: $10^x = 1000$.

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

12. Solve the inequality: $\text{ln}(x) < 0$.

    $0 < x < 1$. Natural log is negative when input is between 0 and 1.

13. Simplify: $\text{log}_b(b^x)$.

    $x$. Logarithm and exponential cancel each other as inverse functions.

14. Simplify: $\text{log}_b(b)$.

    1. Logarithm of the base itself always equals 1.

15. Find the solution: $\text{log}_3(x) = 0$.

    $x = 1$. Convert to exponential: $3^0 = 1$, so $x = 1$.

16. What is the inverse of $y = 10^x$?

    $y = \text{log}_{10}(x)$. Common logarithm is inverse of base-10 exponential function.

17. State the property: $\text{log}_b(x^r)$.

    $r \text{log}_b(x)$. Power rule: exponent becomes coefficient in logarithm.

18. Define the natural logarithm function.

    $\text{ln}(x) = \text{log}_e(x)$. Natural log uses base $e$ (Euler's number) as the base.

19. What is the domain of $\text{log}_b(x)$?

    $x > 0$. Logarithm only defined for positive real numbers.

20. What is the inverse of $\text{ln}(x)$?

    $e^x$. Natural logarithm and exponential function are inverses.

21. Solve for $x$: $5^x = 25$.

    $x = 2$. Since $25 = 5^2$, we have $x = 2$.

22. What is the base of the common logarithm?

    10. Common logarithm uses base 10 by convention.

23. What is the value of $\text{ln}(1)$?

    0. Natural logarithm of 1 is always zero.

24. What is the range of the natural logarithm function?

    All real numbers. Logarithm function outputs all real values for positive inputs.

25. What is the inverse of an exponential function $y = b^x$?

    $x = \text{log}_b(y)$. Exponential and logarithmic functions are inverse operations.

26. What does the equation $b^{\text{log}_b(x)} = x$ represent?

    Identity property of logarithms. Exponential and logarithm cancel each other as inverse functions.

27. Simplify: $\text{log}_a(1)$.

    0. Logarithm of 1 equals zero for any base.

28. State the base of the natural logarithm.

    $e$. Natural logarithm uses Euler's number $e ≈ 2.718$ as base.

29. State the property: $\text{log}_b(xy)$.

    $\text{log}_b(x) + \text{log}_b(y)$. Product rule: logarithm of product equals sum of logarithms.

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

    $f(x) = ab^x$ where $a \neq 0$, $b > 0$, $b \neq 1$. Standard form where $a$ is initial value and $b$ is growth/decay factor.

31. Solve the inequality: $3^x > 9$.

    $x > 2$. Since $9 = 3^2$, we need $x > 2$.

32. Convert to exponential form: $\text{log}_b(x) = y$.

    $b^y = x$. Definition of logarithm as the inverse of exponential function.

33. What is the range of $f(x) = b^x$ for $b > 1$?

    $(0, \text{inf})$. Exponential functions with $b > 1$ produce all positive outputs.

34. What is the range of the function $f(x) = \text{log}_b(x)$?

    All real numbers. Logarithmic functions output all real values.

35. State the property: $\text{log}_b(\frac{x}{y})$.

    $\text{log}_b(x) - \text{log}_b(y)$. Quotient rule: logarithm of quotient equals difference of logarithms.

36. Identify the property: $b^{\text{log}_b(x)}$.

    $x$. Identity property: base raised to its own logarithm.

37. What is $\text{log}_b(1)$ for any base $b$?

    0. Any base raised to power 0 equals 1.

38. Solve for $x$: $\text{log}_2(x) = 3$.

    $x = 8$. Convert to exponential form: $2^3 = 8$, so $x = 8$.

39. State the formula for the natural exponential function.

    $f(x) = e^x$. Uses Euler's number $e$ as the base for the exponential function.

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