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ACT Math

ACT Math Help: Logarithmic Functions

Review real example questions for Logarithmic Functions in ACT Math.

Question 1

What is the value of xxx that satisfies log⁡3(x+2)+log⁡3(x−4)=3\log_3(x + 2) + \log_3(x - 4) = 3log3​(x+2)+log3​(x−4)=3?

  1. 555
  2. 777
  3. 999
  4. 111111
Explanation: This is a logarithms question testing the product rule and extraneous solution detection. Choice B (7) is correct — apply the log product rule: log₃(x + 2) + log₃(x − 4) = log₃((x + 2)(x − 4)) = 3. Convert to exponential form: (x + 2)(x − 4) = 3³ = 27. Expand: x² − 2x − 8 = 27 → x² − 2x − 35 = 0 → (x − 7)(x + 5) = 0 → x = 7 or x = −5. Check: x = −5 makes log₃(−5 + 2) = log₃(−3), which is undefined (can't take log of a negative). So x = 7 is the only valid solution. Choice A (5) comes from a factoring error: perhaps solving x² − 2x − 35 = 0 as (x − 5)(x + 7) = 0. Choice C (9) comes from treating each log separately: log₃(x + 2) = 3 → x + 2 = 27 → x = 25... or log₃(x − 4) = 3 → x − 4 = 27 → x = 31. Choice D (11) comes from adding: (x + 2) + (x − 4) = 27 → 2x − 2 = 27 → x = 14.5, rounding or computing differently. Pro tip: After applying the log product rule, you'll get a quadratic. It will typically have two roots — always check BOTH in the original equation. A root that produces a negative or zero argument for any logarithm is extraneous and must be discarded.

Question 2

A worksheet asks you to simplify log⁡(2)+log⁡(50)\log(2)+\log(50)log(2)+log(50) (base 10). Which single logarithm is equivalent?

  1. log⁡(52)\log(52)log(52)
  2. log⁡ ⁣(250)\log\!\left(\dfrac{2}{50}\right)log(502​)
  3. log⁡(100)\log(100)log(100)
  4. log⁡(2⋅50)⋅log⁡(10)\log(2\cdot 50)\cdot\log(10)log(2⋅50)⋅log(10)
Explanation: This problem uses the logarithm product property. The product property states that log_a(x) + log_a(y) = log_a(xy). Applying this property to log(2) + log(50), we get log(2) + log(50) = log(2 × 50) = log(100). Choice A incorrectly adds the arguments instead of multiplying them.

Question 3

A student wants a single logarithm equivalent to ln⁡(12)−ln⁡(3)\ln(12)-\ln(3)ln(12)−ln(3). Which expression is equivalent?

  1. ln⁡(9)\ln(9)ln(9)
  2. ln⁡(4)\ln(4)ln(4)
  3. ln⁡(15)\ln(15)ln(15)
  4. ln⁡ ⁣(312)\ln\!\left(\dfrac{3}{12}\right)ln(123​)
Explanation: This problem uses the logarithm quotient property. The quotient property states that ln⁡(a)−ln⁡(b)=ln⁡(a/b)\ln(a) - \ln(b) = \ln(a/b)ln(a)−ln(b)=ln(a/b). Applying this property to ln⁡(12)−ln⁡(3)\ln(12) - \ln(3)ln(12)−ln(3), we get ln⁡(12)−ln⁡(3)=ln⁡(12/3)=ln⁡(4)\ln(12) - \ln(3) = \ln(12/3) = \ln(4)ln(12)−ln(3)=ln(12/3)=ln(4). Choice A incorrectly keeps the arguments separate, while choice D incorrectly reverses the fraction.

Question 4

What is log⁡10(1000)\log_{10}(1000)log10​(1000)?

  1. 3
  2. 2
  3. 10
  4. 1
Explanation: To evaluate this logarithm, we need to find what power 10 must be raised to get 1000. The logarithm property states that log_a(b) = c means a^c = b. We can rewrite 1000 as 10^3, so log₁₀(1000) = log₁₀(10^3). Using the power rule for logarithms, log_a(x^n) = n·log_a(x), we get 3·log₁₀(10) = 3·1 = 3.

Question 5

Evaluate log⁡3(81)\log_3(81)log3​(81).

  1. 4
  2. 3
  3. 2
  4. 5
Explanation: To evaluate this logarithm, we need to find what power 3 must be raised to get 81. The logarithm property states that log⁡a(b)=c\log_a(b) = cloga​(b)=c means ac=ba^c = bac=b. We can rewrite 81 as 343^434 (since 3⋅3⋅3⋅3=813 \cdot 3 \cdot 3 \cdot 3 = 813⋅3⋅3⋅3=81). Therefore, log⁡3(81)=log⁡3(34)\log_3(81) = \log_3(3^4)log3​(81)=log3​(34). Using the power rule, this equals 4⋅log⁡3(3)=4⋅1=44 \cdot \log_3(3) = 4 \cdot 1 = 44⋅log3​(3)=4⋅1=4.

Question 6

A finance model uses natural logs. What is ln⁡(e5)\ln(e^5)ln(e5)?

  1. e5e^5e5
  2. 5e5e5e
  3. 555
  4. 15\dfrac{1}{5}51​
Explanation: This problem uses the property that natural logarithm and exponential are inverse functions. The key property is that ln(e^x) = x for any real number x. Applying this property directly to ln(e^5), we get ln(e^5) = 5. Choice A incorrectly leaves the expression unsimplified, while choice B incorrectly multiplies by e.

Question 7

A calculator app uses base-10 logs. If log⁡(x)=−2\log(x)= -2log(x)=−2, what is the value of xxx?

  1. −2-2−2
  2. 10−210^{-2}10−2
  3. 222
  4. −102-10^{2}−102
Explanation: Given log(x) = -2 (base 10 implied), we need to find x. Using the definition log₁₀(x) = -2 means 10^(-2) = x. Therefore, x = 10^(-2) = 1/10² = 1/100 = 0.01. Choice A gives just -2, which is the logarithm value, not x itself.

Question 8

A student simplifies log⁡7(49)\log_7(49)log7​(49). What is the value of log⁡7(49)\log_7(49)log7​(49)?

  1. 12\dfrac{1}{2}21​
  2. 222
  3. 777
  4. 494949
Explanation: This problem uses the fundamental relationship between logarithms and exponentials. The equation log_7(49) asks "to what power must we raise 7 to get 49?" Since 7^2 = 49, we have log_7(49) = 2. Choice A incorrectly gives the reciprocal, while choice C gives the base instead of the exponent.

Question 9

A student simplifies log⁡5(125)\log_5(125)log5​(125). What is the value of log⁡5(125)\log_5(125)log5​(125)?

  1. 222
  2. 333
  3. 555
  4. 125125125
Explanation: This problem uses the fundamental relationship between logarithms and exponentials. The equation log_5(125) asks "to what power must we raise 5 to get 125?" Since 5^3 = 125, we have log_5(125) = 3. Choice D incorrectly gives the argument of the logarithm rather than the exponent needed.

Question 10

What is the value of xxx if log⁡(x)=2\log(x) = 2log(x)=2?

  1. 100
  2. 10
  3. 20
  4. 200
Explanation: To solve this equation, we need to convert from logarithmic to exponential form. The equation log(x) = 2 means "10 raised to what power equals x?" Since log without a specified base typically means log₁₀, we have 10² = x. Therefore, x = 100. Choice B would give 10¹ = 10, which doesn't satisfy the original equation.