Logarithmic Functions - ACT Math

1. Express $b^{\text{log}_b(x)}$ in simplest form.

   $b^{\text{log}_b(x)} = x$. The exponential and logarithm with same base cancel out.

2. What is $ $\log_{10}$(1000)$?

   $3$. Since $10^3 = 1000$, we have $\log_{10}(1000) = 3$.

3. What is $\ln(e^5)$?

   $5$. Natural log and exponential with base $e$ are inverse functions.

4. Solve for $x$: $ $\log_2$(x)=5$.

   $x=32$. Convert to exponential form: $x = 2^5 = 32$.

5. What restriction on the input $x$ is required for $ $\log_b$(x)$ to be defined over the reals?

   $x>0$. Input must be positive since logarithms of non-positive numbers are undefined in reals.

6. What is the value of $\text{log}_e(e^3)$?

   $3$. Natural logarithm and exponential with same base cancel.

7. What vertical asymptote does $f(x)=\log_b(x)$ have?

   Vertical asymptote: $x=0$. Function approaches negative infinity as $x$ approaches 0 from the right.

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

   $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. Quotient inside log equals difference of individual logs.

9. Evaluate: $\text{log}_2(32)$.

   $5$. Since $2^5 = 32$, the logarithm equals 5.

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

    $\log_b(b)=1$. Base raised to power 1 equals itself, so $\log_b(b) = 1$.

11. What is the range of the logarithmic function $f(x)=\log_b(x)$?

    Range: all real numbers. Logarithm can output any real value as input varies over positive reals.

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

    $\log_b(1)=0$. Any base raised to power 0 equals 1, so $\log_b(1) = 0$.

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

    $3$. Since $2^3 = 8$, we have $\log_2(8) = 3$.

14. What is the meaning of $\ln(x)$?

    $\ln(x)=\log_e(x)$. Natural logarithm uses base $e$ (Euler's number).

15. What is the domain of the logarithmic function $f(x)=\log_b(x)$?

    Domain: $x>0$. Input must be positive for logarithm to be defined in real numbers.

16. Evaluate: $\text{log}_5(125)$.

    $3$. Since $5^3 = 125$, the logarithm equals 3.

17. What is the change-of-base formula for $\log_b(x)$ using base $10$?

    $\log_b(x)=\frac{\log(x)}{\log(b)}$. Converts any base logarithm to common logarithm (base 10).

18. What is the domain of $f(x)=\log_b(ax+c)$ in one inequality (assume $a\neq 0$)?

    Domain: $ax+c>0$. Argument must be positive, so $ax + c > 0$ determines the domain.

19. What is the power rule for logarithms: $log_b(M^k)$ equals what?

    $\log_b(M^k)=k\log_b(M)$. Power inside log becomes coefficient multiplying the log.

20. If $\text{log}_5(x) = 0$, what is $x$?

    $x = 1$. Any base raised to the zero power equals 1.

21. Evaluate: $\text{log}_4(64)$.

    $3$. Since $4^3 = 64$, the logarithm equals 3.

22. Evaluate: $\text{ln}(1)$.

    $0$. Natural logarithm of 1 always equals zero.

23. If $a = \text{log}_b(x)$, express $x$ in terms of $b$ and $a$.

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

24. Convert $\text{ln}(x) = 2$ to exponential form.

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

25. Identify the property: $n \times \text{log}_b(x) = \text{log}_b(x^n)$.

    Power Property. Used when raising logarithmic arguments to powers.

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

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

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

    $\text{log}_b(x) = \frac{\text{log}_k(x)}{\text{log}_k(b)}$. Allows conversion between different logarithmic bases.

28. Find the domain of $f(x)=\log_7(2x+3)$ as an inequality in $x$.

    Domain: $x>\frac{-3}{2}$. Argument $2x + 3$ must be positive: $2x + 3 > 0$ gives $x > -\frac{3}{2}$.

29. Solve for $x$: $ $\log_3$(x)=\log_3(27)$.

    $x=27$. If logarithms are equal, their arguments must be equal.

30. Express $\text{log}_b(\frac{x}{y})$ using properties of logarithms.

    $\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)$. The logarithm of a quotient equals the difference of logarithms.

31. Solve for $x$: $ $\log_4$(x-1)=2$.

    $x=17$. Convert to exponential: $x - 1 = 4^2 = 16$, so $x = 17$.

32. What is the base of natural logarithms?

    The base is $e$, where $e \thickapprox 2.718$. Natural logarithms use Euler's number as the base.

33. Solve for $x$: $\text{log}_x(16) = 2$.

    $x = 4$. Convert to exponential form: $x^2 = 16$, so $x = 4$.

34. Find the domain of $f(x)=\log_2(x-5)$ as an inequality in $x$.

    Domain: $x>5$. Argument $x - 5$ must be positive for logarithm to be defined.

35. Express $\text{log}_b(xy)$ using properties of logarithms.

    $\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)$. The logarithm of a product equals the sum of logarithms.

36. What is the change-of-base formula for $\log_b(x)$ using base $e$?

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

37. Simplify: $\text{ln}(e^5)$.

    $5$. Natural logarithm and exponential cancel each other.

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

    $\log_b(M^k)=k\log_b(M)$. Power inside log becomes coefficient multiplying the log.

39. Simplify using properties: $\text{log}_b(b^{-3})$.

    $-3$. The logarithm and exponential with same base cancel.

40. Solve for $x$: $ $\log_2$(x)+\log_2(4)=6$.

    $x=16$. Use product rule: $\log_2(4x) = 6$, so $4x = 2^6 = 64$, giving $x = 16$.

41. What is the change-of-base formula for $ $\log_b$(x)$ using base $e$?

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

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

    $\text{log}_b(b) = 1$. A base raised to the first power equals itself.

43. Express $\text{log}_b(x^n)$ using properties of logarithms.

    $\text{log}_b(x^n) = n \times \text{log}_b(x)$. The exponent can be brought down as a coefficient.

44. What is $ $\log_3$\left(\frac{1}{9}\right)$?

    $-2$. Since $3^{-2} = \frac{1}{9}$, we have $\log_3\left(\frac{1}{9}\right) = -2$.

45. For $0<b<1$, is $f(x)=\log_b(x)$ increasing or decreasing on $x>0$?

    Decreasing on $x>0$. For bases between 0 and 1, larger inputs yield smaller logarithm values.

46. Simplify $ $\log_5$(125x)$ as a sum of logs.

    $\log_5(125x)=3+\log_5(x)$. Use product rule: $\log_5(125) + \log_5(x) = 3 + \log_5(x)$ since $5^3 = 125$.

47. What is the inverse relationship between $b^x$ and $ $\log_b$(x)$?

    $b^{\log_b(x)}=x$ and $\log_b(b^x)=x$. Exponential and logarithm functions are inverse operations that cancel each other.

48. What restriction on the base $b$ is required for $ $\log_b$(x)$ to be a logarithm?

    $b>0$ and $b\neq 1$. Base must be positive and not equal to 1 for logarithm to be well-defined.

49. What is $ $\log_b$($b^k$)$ when $b>0$ and $b\neq 1$?

    $\log_b(b^k)=k$. Logarithm of a power of the base equals the exponent.

50. What is the meaning of $ log(x)$ on the ACT when no base is written?

    $\log(x)=\log_{10}(x)$. When no base is specified, logarithm assumes base 10 (common logarithm).

51. Evaluate: $\text{ln}(e)$.

    $1$. Natural logarithm of its base equals 1.

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

    $\log_b(1)=0$. Any base raised to power 0 equals 1, so $\log_b(1) = 0$.

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

    $\log_b\left(\frac{M}{N}\right)=\log_b(M)-\log_b(N)$. Quotient inside log equals difference of individual logs.

54. Find $\text{log}_{10}(1000)$.

    $3$. Since $10^3 = 1000$, the common logarithm equals 3.

55. Condense $\log_b(x)+\log_b(y)-\log_b(z)$ into one logarithm.

    $\log_b\left(\frac{xy}{z}\right)$. Use product rule backwards, then quotient rule backwards.

56. What is the inverse function of $y = b^x$?

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

57. Simplify: $\text{log}_b(1)$.

    $\text{log}_b(1) = 0$. Any base raised to the zero power equals 1.

58. Solve for $x$: $log_4(x-1)=2$.

    $x=17$. Convert to exponential: $x - 1 = 4^2 = 16$, so $x = 17$.

59. If $\text{log}_b(x) = 3$, express $x$ in terms of $b$.

    $x = b^3$. Convert from logarithmic to exponential form.

60. What vertical asymptote does $f(x)=\log_b(x-h)+k$ have?

    Vertical asymptote: $x=h$. Horizontal shift moves the vertical asymptote to $x = h$.

61. Solve for $x$: $ $\log_5$(x)=0$.

    $x=1$. Since $\log_5(1) = 0$, we have $x = 1$.

62. If $\text{log}_b(10) = 1.5$, what is $\text{log}_b(100)$?

    $3$. Use power property: $\text{log}_b(100) = \text{log}_b(10^2) = 2 \times 1.5$.

63. Simplify $ $\log_2$($16x^3$)$ using log rules.

    $\log_2(16x^3)=4+3\log_2(x)$. Use product and power rules: $\log_2(16) + \log_2(x^3) = 4 + 3\log_2(x)$.

64. What transformation gives the domain condition for $f(x)=\log_b(x-h)+k$?

    Domain: $x>h$. Horizontal shift by $h$ units moves the domain restriction accordingly.

65. Solve for $x$: $10^{\text{log}(x)} = 100$.

    $x = 100$. Since $10^{\text{log}(x)} = x$, we have $x = 100$.

66. Solve for $x$: $\text{log}_7(x) = -1$.

    $x = \frac{1}{7}$. Convert to exponential: $7^{-1} = \frac{1}{7}$.

67. If $\text{log}_b(x) = 2$ and $\text{log}_b(y) = 3$, find $\text{log}_b(xy)$.

    $5$. Use product property: $\text{log}_b(xy) = 2 + 3 = 5$.

68. Evaluate: $\text{log}_2(1)$.

    $0$. Any logarithm of 1 equals zero.

69. State the logarithmic identity for $\text{log}_b(b^x)$.

    $\text{log}_b(b^x) = x$. The base and its logarithm cancel each other out.

70. What is the meaning of $ ln(x)$?

    $\ln(x)=\log_e(x)$. Natural logarithm uses base $e$ (Euler's number).

71. What is the definition of $ $\log_b$(x)$ in terms of an exponential equation?

    $\log_b(x)=y\iff b^y=x$. Converts between logarithmic and exponential forms using equivalent definitions.

72. If $\text{log}_b(2) = 0.301$, find $\text{log}_b(8)$.

    $0.903$. Use power property: $\text{log}_b(8) = \text{log}_b(2^3) = 3 \times 0.301$.

73. Identify the property: $\text{log}_b(x) + \text{log}_b(y) = \text{log}_b(xy)$.

    Product Property. Used when multiplying arguments inside logarithms.

74. What is the value of $\text{log}_a(a^x)$?

    $\text{log}_a(a^x) = x$. The logarithm and exponential with same base cancel.

75. What is the product rule for logarithms: $ $\log_b$(MN)$ equals what?

    $\log_b(MN)=\log_b(M)+\log_b(N)$. Product inside log equals sum of individual logs.

76. Solve for $x$: $ $\log_2$(x)+\log_2(x)=6$.

    $x=8$. Simplify to $\log_2(x^2) = 6$, so $x^2 = 2^6 = 64$, giving $x = 8$.

77. Expand $ $\log_3$\left(\frac{x^2}{y}\right)$ using log rules.

    $\log_3\left(\frac{x^2}{y}\right)=2\log_3(x)-\log_3(y)$. Apply quotient rule, then power rule to expand completely.

78. What is the $x$-intercept of $f(x)=\log_b(x)$ for any valid base $b$?

    $x$-intercept: $(1,0)$. Graph crosses x-axis where $\log_b(x) = 0$, which occurs at $x = 1$.

79. What is $b^{\log_b(k)}$ when $k>0$?

    $b^{\log_b(k)}=k$. Base raised to its own logarithm equals the argument.

80. What is the definition of a logarithm?

    If $b^x = y$, then $\text{log}_b(y) = x$. The logarithm is the inverse operation of exponentiation.

81. Identify the property: $\text{log}_b(x) - \text{log}_b(y) = \text{log}_b(\frac{x}{y})$.

    Quotient Property. Used when dividing arguments inside logarithms.

82. Solve for $x$: $\text{log}_3(x) = 4$.

    $x = 81$. Convert to exponential form: $3^4 = 81$.

83. Solve for $x$: $ $\log_3$(x)-\log_3(9)=1$.

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

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

    $7$. Exponential and natural log are inverse functions that cancel.

85. For $b>1$, is $f(x)=\log_b(x)$ increasing or decreasing on $x>0$?

    Increasing on $x>0$. For bases greater than 1, larger inputs yield larger logarithm values.

86. Find the vertical asymptote of $f(x)=\log_3(x+4)$.

    Vertical asymptote: $x=-4$. Vertical asymptote occurs where the argument equals zero: $x + 4 = 0$.

87. Evaluate: $\text{log}_9(81)$.

    $2$. Since $9^2 = 81$, the logarithm equals 2.

88. Condense $2\log_b(x)-3\log_b(y)$ into one logarithm.

    $\log_b\left(\frac{x^2}{y^3}\right)$. Apply power rule backwards, then quotient rule backwards.

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