Exponents, Logarithms, and Their Inverse Relationship - Algebra 2

1. What is the quotient rule for logarithms: $\log_b\left(\frac{M}{N}\right)$ equals what (for $M>0$, $N>0$)?

   $\log_b(M)-\log_b(N)$. Logarithm of a quotient equals difference of logarithms.

2. What is the horizontal asymptote of $y=b^x$ for $b>0$ and $b\ne 1$?

   $y=0$. Exponential function approaches 0 as $x$ approaches negative infinity.

3. What is the inverse statement that matches $a^b=c$ using logarithms (with $a>0$, $a\ne 1$, $c>0$)?

   $\log_a(c)=b$. Logarithmic form: the exponent that gives $c$ when $a$ is raised to it.

4. What is the exact value of $\log_7(\frac{1}{49})$?

   $-2$. Since $7^{-2} = \frac{1}{49}$.

5. What is the exact solution of $\log_2(x-1)=3$?

   $x=9$. Since $2^3 = 8$, so $x - 1 = 8$.

6. What is the value of $\ln(e^x)$ for any real $x$?

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

7. What is the exact value of $\log_{16}(\frac{1}{2})$?

   $-\frac{1}{4}$. Since $16^{-1/4} = (2^4)^{-1/4} = 2^{-1} = \frac{1}{2}$.

8. What is the exact solution of $\log_4(x)=0$?

   $x=1$. Since $4^0 = 1$.

9. What is the exact solution of $\log_{10}(x)= -1$?

   $x=0.1$. Since $10^{-1} = 0.1$.

10. What is the range of $f(x)=b^x$ for $b>0$ and $b\ne 1$?

    $(0,\infty)$. Exponential functions have positive outputs for all real inputs.

11. What is the exact value of $\log_{27}(9)$?

    $\frac{2}{3}$. Since $27^{2/3} = (3^3)^{2/3} = 3^2 = 9$.

12. What is the exact solution of $3^x=\frac{1}{27}$?

    $x=-3$. Since $3^{-3} = \frac{1}{27}$.

13. What is the value of $\log_b(b^x)$ for any real $x$ and valid base $b$?

    $x$. Logarithmic and exponential functions are inverses.

14. What is the value of $10^{\log(x)}$ for $x>0$?

    $x$. Common logarithm and exponential base 10 are inverse functions.

15. What is the exact solution of $10^x=0.001$?

    $x=-3$. Since $10^{-3} = 0.001$.

16. What is $\log_2(8)$?

    $3$. Since $2^3 = 8$.

17. What is the exact solution of $4^x=2$?

    $x=\frac{1}{2}$. Since $4^{1/2} = 2$.

18. What is the exponential statement that matches $\log_a(c)=b$ (with $a>0$, $a\ne 1$, $c>0$)?

    $a^b=c$. Exponential form: $a$ raised to the power $b$ equals $c$.

19. What is $\ln(1)$?

    $0$. Any base raised to power 0 equals 1.

20. What is $\log_{\frac{1}{2}}(4)$?

    $-2$. Since $(\frac{1}{2})^{-2} = 2^2 = 4$.

21. What are the allowed base values for $\log_b(x)$ in the real number system?

    $b>0$ and $b\ne 1$. Base must be positive and not equal to 1 for valid logarithms.

22. What is the exact value of $\log_2(\sqrt{32})$?

    $\frac{5}{2}$. Using power rule: $\log_2(32^{1/2}) = \frac{1}{2} \cdot 5 = \frac{5}{2}$.

23. What is the domain of $\log_b(x)$ for real outputs (assume $b>0$, $b\ne 1$)?

    $x>0$. Logarithms are only defined for positive arguments.

24. What is $\log_4(2)$?

    $\frac{1}{2}$. Since $4^{1/2} = \sqrt{4} = 2$.

25. What is the exact solution of $\ln(x)=\ln(12)$?

    $x=12$. Equal natural logarithms means equal arguments.

26. What is $\log_3(\frac{1}{9})$?

    $-2$. Since $3^{-2} = \frac{1}{9}$.

27. What is the exact solution of $5^{2x}=125$?

    $x=\frac{3}{2}$. Since $5^3 = 125$, so $2x = 3$.

28. What is the product rule for logarithms: $\log_b(MN)$ equals what (for $M>0$, $N>0$)?

    $\log_b(M)+\log_b(N)$. Logarithm of a product equals sum of logarithms.

29. What is the exact solution of $e^x=\frac{1}{e^5}$?

    $x=-5$. Since $e^{-5} = \frac{1}{e^5}$.

30. What is $\log_7(7)$?

    $1$. Any base raised to power 1 equals itself.

31. What is the exact solution of $\log_2(x)=\log_2(7)$?

    $x=7$. Equal logarithms with same base means equal arguments.

32. What is the change-of-base formula for $\log_b(x)$ using natural logs?

    $\log_b(x)=\frac{\ln(x)}{\ln(b)}$. Converts logarithm to any base using natural logarithms.

33. What is the exact value of $\log_5(125)-\log_5(5)$?

    $2$. Using quotient rule: $\log_5(\frac{125}{5}) = \log_5(25) = 2$.

34. What is the change-of-base formula for $\log_b(x)$ using common logs?

    $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts logarithm to any base using common logarithms.

35. What is the value of $b^{\log_b(x)}$ for $x>0$ and valid base $b$?

    $x$. Exponential and logarithmic functions are inverses.

36. What is $e^{\ln(7)}$?

    $7$. Exponential and natural logarithm are inverse functions.

37. What is the exact value of $\log_3(\frac{81}{3})$?

    $3$. Using quotient rule: $\log_3(\frac{81}{3}) = \log_3(27) = 3$.

38. What is the exact value of $\log_{\frac{1}{4}}(2)$?

    $-\frac{1}{2}$. Since $(\frac{1}{4})^{-1/2} = 4^{1/2} = 2$.

39. What is the value of $e^{\ln(x)}$ for $x>0$?

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

40. What is the exact value of $\log_8(4)$?

    $\frac{2}{3}$. Since $8^{2/3} = (2^3)^{2/3} = 2^2 = 4$.

41. What is $\ln(e^3)$?

    $3$. Natural logarithm and exponential are inverse functions.

42. What is the key condition to take $\log_b$ of both sides of an equation in real numbers?

    Both sides must be $>0$. Logarithm domain requires positive arguments for real outputs.

43. What is the inverse function of $f(x)=b^x$ (with $b>0$, $b\ne 1$) written as a log?

    $f^{-1}(x)=\log_b(x)$. Exponential and logarithmic functions are inverses of each other.

44. What is $\log_5(1)$?

    $0$. Any base raised to power 0 equals 1.

45. What is the value of $\log_b(1)$ for any valid base $b$?

    $0$. Since $b^0 = 1$ for any valid base $b$.

46. What is the exact solution of $2^x=16$?

    $x=4$. Since $2^4 = 16$.

47. What is the exact solution of $\log_5(x)=3$?

    $x=125$. Since $5^3 = 125$.

48. What is the exact solution of $\log_3(x)-\log_3(9)=1$?

    $x=27$. Using quotient rule: $\log_3(\frac{x}{9}) = 1$, so $\frac{x}{9} = 3$.

49. What is $\log_2(32)$?

    $5$. Since $2^5 = 32$.

50. What is the power rule for logarithms: $\log_b(M^k)$ equals what (for $M>0$)?

    $k\log_b(M)$. Logarithm of a power equals exponent times logarithm of base.

51. What is the vertical asymptote of $y=\log_b(x)$ for any valid base $b$?

    $x=0$. Logarithm approaches negative infinity as $x$ approaches 0 from the right.

52. What is the exact solution of $\log_2(x)+\log_2(4)=5$?

    $x=8$. Using product rule: $\log_2(4x) = 5$, so $4x = 32$.

53. What is the value of $\log_b(b)$ for any valid base $b$?

    $1$. Since $b^1 = b$ for any valid base $b$.

54. What is the exact solution of $2^{x+1}=32$?

    $x=4$. Since $2^5 = 32$, so $x + 1 = 5$.

55. What is $\log_{10}(0.01)$?

    $-2$. Since $10^{-2} = 0.01$.

56. What is the range of $g(x)=\log_b(x)$ for a valid base $b$?

    $(-\infty,\infty)$. Logarithmic functions output all real numbers for positive inputs.

57. What is the exact value of $\log_9(3)$?

    $\frac{1}{2}$. Since $9^{1/2} = 3$, so $\log_9(3) = \frac{1}{2}$.

58. What is the exact solution of $\ln(x)=2$?

    $x=e^2$. Converting logarithmic to exponential form: $e^2 = x$.

59. What is the exact solution of $\log_3(x^2)=2$ with $x>0$?

    $x=3$. Using power rule: $2\log_3(x) = 2$, so $\log_3(x) = 1$.

60. What is the exact solution of $\log_2(x)=-4$?

    $x=\frac{1}{16}$. Since $2^{-4} = \frac{1}{16}$.

61. What is the meaning of $\log_b(x)$ in words (assume $b>0$, $b\ne 1$, $x>0$)?

    The exponent $y$ such that $b^y=x$. The logarithm asks: what power makes $b^y = x$?

62. What is the exact solution of $\log_3(x)=\frac{1}{2}$?

    $x=\sqrt{3}$. Since $3^{1/2} = \sqrt{3}$.

63. What is the inverse function of $g(x)=\log_b(x)$ (with valid base $b$) written exponentially?

    $g^{-1}(x)=b^x$. Logarithmic and exponential functions are inverses of each other.

64. What is the exact value of $\log_2(8\cdot 4)$ using log properties?

    $5$. Using product rule: $\log_2(8) + \log_2(4) = 3 + 2 = 5$.

65. What is the value of $\log_{10}(10^x)$ for any real $x$?

    $x$. Common logarithm and exponential base 10 are inverse functions.

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