Logarithmic Functions - ACT Math

1. Evaluate $\log_{2}(32)$.
   Answer: $5$ We need to find what power of 2 equals 32, using $\log_a$(b) = c means $a^c$ = b. Since 32 = 2⁵ (because 2×2×2×2×2 = 32), we have 2⁵ = 32. Therefore, log₂(32) = 5. Choice D incorrectly gives 2³² which would be the result of raising 2 to the power of 32, not the logarithm value.
2. A student is converting between logarithmic and exponential forms. Which equation is equivalent to $\log_3(81)=4$?
   Answer: $3^4=81$ This problem requires converting from logarithmic to exponential form. The logarithmic equation $\log_a$(b) = c is equivalent to the exponential equation $a^c$ = b. For $\log_3$(81) = 4, the base is 3, the result is 4, and the argument is 81. Therefore, the equivalent exponential form is $3^4$ = 81. Choice B incorrectly switches the base and exponent.
3. A solution's acidity is measured on the pH scale, where $\text{pH}=-\log_{10}([H^+])$. If $[H^+]=10^{-5}$, what is the pH?
   Answer: $5$ To find pH, we need to apply the given formula pH = -log₁₀([H⁺]) with [H⁺] = 10⁻⁵. The logarithm property we need is $\log_a$($a^x$) = x, which tells us that log₁₀(10⁻⁵) = -5. Substituting into the pH formula: pH = -log₁₀(10⁻⁵) = -(-5) = 5. Choice C incorrectly gives the hydrogen ion concentration itself rather than the pH value.
4. A student wants a single logarithm equivalent to $\ln(12)-\ln(3)$. Which expression is equivalent?
   Answer: $\ln(4)$ This problem uses the logarithm quotient property. The quotient property states that $\ln(a) - \ln(b) = \ln(a/b)$. Applying this property to $\ln(12) - \ln(3)$, we get $\ln(12) - \ln(3) = \ln(12/3) = \ln(4)$. Choice A incorrectly keeps the arguments separate, while choice D incorrectly reverses the fraction.
5. $\log_2(16)$ is equivalent to which of the following?
   Answer: 4 This problem uses the definition of logarithms with base 2. The expression $log_2(16)$ asks "To what power must we raise 2 to get 16?" Since $2^4 = 2 \times 2 \times 2 \times 2 = 16$, we have $log_2(16) = 4$. Choice B would be incorrect as $2^3 = 8$, not 16.
6. A student solves the equation $\ln(x)=0$. What is the value of $x$?
   Answer: $1$ This problem uses the fundamental property that ln(1) = 0 and the inverse relationship between natural logarithm and exponential functions. The equation ln(x) = 0 means that $e^0$ = x. Since $e^0$ = 1 for any base, we have x = 1. Choice A incorrectly gives the exponent value, while choice C gives the base of natural logarithm.
7. Which equation is equivalent to $\log_{3}(81)=c$?
   Answer: $3^{c}=81$ The equation log₃(81) = c uses the definition $\log_a$(b) = c means $a^c$ = b. Converting from logarithmic to exponential form: if log₃(81) = c, then $3^c$ = 81. This directly translates the logarithmic equation into its equivalent exponential form. Choice A incorrectly reverses the base and exponent, giving c³ = 81.
8. Solve for $x$ if $\ln(x) = 1$.
   Answer: $e$ To solve this equation, we need to convert from logarithmic to exponential form. The logarithm property states that $\log_a$(b) = c means $a^c$ = b. Given ln(x) = 1, where ln means $\log_e$, we convert to exponential form: $e^1$ = x. Therefore, x = e.
9. What is $\log_{10}(1000)$?
   Answer: 3 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.
10. Which expression is equivalent to $\log(\frac{x}{y})$?
    Answer: $\log(x) - \log(y)$ This problem uses the quotient rule for logarithms. The quotient rule states that log(x/y) = log(x) - log(y). This property allows us to separate the logarithm of a fraction into the difference of two logarithms. Choice B incorrectly suggests adding the logarithms, which would apply to a product, not a quotient.
11. A student evaluates $\log_9(3)$. What is the value of $\log_9(3)$?
    Answer: $\dfrac{1}{2}$ This problem uses the relationship between logarithms with different bases that are powers of each other. Since 9 = $3^2$, we need to find what power of 9 equals 3. This means $9^x$ = 3, or $(3^2)^x$ = $3^1$, which gives us $3^{2x}$ = $3^1$. Therefore, 2x = 1, so x = 1/2. Choice B incorrectly gives the reciprocal relationship.
12. A calculus student uses log rules to rewrite $\ln(\sqrt{e^{10}})$. What is its value?
    Answer: $5$ This problem combines the square root property with the logarithm power property. First, $\sqrt{e^{10}} = (e^{10})^{1/2} = e^{10 \cdot 1/2} = e^5$. Then, using the property that $\ln(e^x) = x$, we have $\ln(e^5) = 5$. The key insight is recognizing that the square root creates a power of 1/2, which brings down the exponent when using the power property.
13. Solve for $x$: $\log_{4}(x)=\frac{1}{2}$.
    Answer: $2$ Given log₄(x) = 1/2, we use the definition $\log_a$(b) = c means $a^c$ = b. This gives us $4^{1/2}$ = x. Since $4^{1/2}$ = √4 = 2, we have x = 2. Choice B incorrectly squares the base instead of taking its square root.
14. Which equation is equivalent to $\log_6(36) = 2$?
    Answer: $6^2 = 36$ To convert from logarithmic to exponential form, we use the fundamental property that $\log_a$(b) = c is equivalent to $a^c$ = b. In the equation log₆(36) = 2, the base is 6, the argument is 36, and the exponent is 2. Converting to exponential form gives us $6^2$ = 36. This can be verified: 6·6 = 36.
15. A lab report states $\log_2(32)=x$. What is the value of $x$?
    Answer: $5$ This problem uses the fundamental relationship between logarithms and exponentials. The equation $\log_2(32) = x$ means that $2^x = 32$. To find x, we need to determine what power of 2 equals 32. Since $2^5 = 32$, we have $x = 5$. Choice D incorrectly gives the argument of the logarithm rather than the exponent.
16. $\log_2(8) + \log_2(4)$ equals what?
    Answer: 5 This problem requires the product rule for logarithms. The property states that $\log_a(x) + \log_a(y) = \log_a(xy)$. First, let's evaluate each term: $ \log_2(8) = 3 $ (since $ $2^3$ = 8 $) and $ \log_2(4) = 2 $ (since $ $2^2$ = 4 $). Adding these together: $ 3 + 2 = 5 $. Alternatively, we could use the product rule: $\log_2(8) + \log_2(4) = \log_2(8·4) = \log_2(32) = 5$.
17. Which equation is equivalent to $\log_{10}(100) = 2$?
    Answer: $10^2 = 100$ To convert from logarithmic to exponential form, we use the fundamental property that $\log_a$(b) = c is equivalent to $a^c$ = b. Given log₁₀(100) = 2, the base is 10, the exponent is 2, and the result is 100. Therefore, the equivalent exponential equation is $10^2$ = 100. Choice B incorrectly switches the base and exponent positions.
18. For a positive constant $a$, simplify $\log_{10}(a^4)$ using the power rule.
    Answer: $4\log_{10}(a)$ To simplify log₁₀(a⁴), we apply the power rule for logarithms. The power rule states that $\log_b$($x^n$) = n·$\log_b$(x). Therefore, log₁₀(a⁴) = 4·log₁₀(a). Choice A incorrectly raises the logarithm itself to the fourth power, while choice D would mean log₁₀(4a) = log₁₀(4) + log₁₀(a).
19. A finance model uses natural logs. What is $\ln(e^5)$?
    Answer: $5$ 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.
20. A student is told that logarithms require positive arguments. Which of the following expressions is not defined in the real numbers?
    Answer: $\log(-10)$ This problem tests understanding of the domain of logarithmic functions. Logarithms are only defined for positive real arguments. Among the choices, log(0.1) is defined since 0.1 > 0, ln(5) is defined since 5 > 0, and log(100) is defined since 100 > 0. However, log(-10) is not defined in the real numbers because -10 < 0.

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