Logarithm Properties - Accuplacer Aaf

1. Which of the following statements about logarithms is always true for positive real numbers $a$, $b$, and valid bases?
   Answer: $\log(a^b) = b \log a$ The power property states $\log(a^b) = b \log a$ is always true. Choice A is false; $\log(a + b) \neq \log a + \log b$. Choice B is false; the correct product property is $\log(ab) = \log a + \log b$. Choice C is false; the correct quotient property is $\log\left(\frac{a}{b}\right) = \log a - \log b$.
2. If $\log_a x = p$ and $\log_a y = q$, then $\log_a\left(\frac{x^2\sqrt{y}}{a}\right)$ equals:
   Answer: $2p + \frac{q}{2} - 1$ Using logarithm properties: $\log_a\left(\frac{x^2\sqrt{y}}{a}\right) = \log_a(x^2\sqrt{y}) - \log_a(a) = \log_a(x^2) + \log_a(\sqrt{y}) - 1 = 2\log_a x + \frac{1}{2}\log_a y - 1 = 2p + \frac{q}{2} - 1$. Choice B incorrectly adds 1 instead of subtracting. Choice C incorrectly combines the first two terms. Choice D incorrectly takes the square root of q outside the logarithm.
3. Given that $\log_3 2 \approx 0.631$, which value is closest to $\log_3 72$?
   Answer: $4.0$ When you encounter logarithm problems with unfamiliar bases, the key strategy is to break down complex expressions using logarithm properties, especially the product rule: $\log_b(xy) = \log_b x + \log_b y$. To find $\log_3 72$, start by factoring 72 into familiar components. Since $72 = 8 \times 9$, we can write: $\log_3 72 = \log_3(8 \times 9) = \log_3 8 + \log_3 9$ Now evaluate each piece. Since $9 = 3^2$, we know $\log_3 9 = 2$. For $\log_3 8$, notice that $8 = 2^3$, so: $\log_3 8 = \log_3(2^3) = 3\log_3 2 = 3(0.631) = 1.893$ Therefore: $\log_3 72 = 1.893 + 2 = 3.893 \approx 4.0$ This confirms answer choice D is correct. Choice A (3.5) likely comes from incorrectly calculating $\log_3 8$ or making arithmetic errors in the final sum. Choice B (5.2) suggests confusion about logarithm properties, possibly adding when you should multiply or using the wrong base conversion. Choice C (4.6) might result from miscalculating $3 \times 0.631$ or incorrectly handling the factorization of 72. Study tip: When working with unfamiliar logarithms, always factor the argument into powers of the base (like $3^2 = 9$) and powers of numbers you're given information about (like $2^3 = 8$). This makes complex logarithms much more manageable using basic logarithm properties.
4. Which expression is equivalent to $\log_3(27x^2) - \log_3(9x)$ for $x > 0$?
   Answer: $\log_3(3x)$ Using the quotient property: $\log_3(27x^2) - \log_3(9x) = \log_3\left(\frac{27x^2}{9x}\right) = \log_3(3x)$. Choice B incorrectly adds the arguments. Choice C equals $\log_3(3x)$ but isn't the simplified form requested. Choice D incorrectly multiplies the arguments.
5. If $\log_2 x + \log_2 y = 5$ and $\log_2 x - \log_2 y = 1$, what is the value of $xy$?
   Answer: 32 Using the product and quotient properties: $\log_2 x + \log_2 y = \log_2(xy) = 5$, so $xy = 2^5 = 32$. We can verify by solving the system: adding the equations gives $2\log_2 x = 6$, so $x = 8$. Subtracting gives $2\log_2 y = 4$, so $y = 4$. Thus $xy = 32$.
6. If $\log_5 a = 2$ and $\log_5 b = 3$, what is the value of $\log_5\left(\frac{a^3}{b^2}\right)$?
   Answer: $0$ Using logarithm properties: $\log_5\left(\frac{a^3}{b^2}\right) = \log_5(a^3) - \log_5(b^2) = 3\log_5 a - 2\log_5 b = 3(2) - 2(3) = 6 - 6 = 0$. Choice A results from incorrectly calculating $2(3) - 3(2)$. Choice C comes from $3 - 2$. Choice D comes from $3(2)$.
7. For positive values of $x$, $y$, and $z$, which expression is equivalent to $\log\left(\frac{x^3y^2}{z^4}\right)$?
   Answer: $3\log x + 2\log y - 4\log z$ When you encounter logarithms of complex expressions involving multiplication, division, and exponents, you need to apply the fundamental logarithm properties to break them down into simpler components. The expression $\log\left(\frac{x^3y^2}{z^4}\right)$ can be simplified using three key logarithm rules: - $\log\left(\frac{a}{b}\right) = \log a - \log b$ (quotient rule) - $\log(ab) = \log a + \log b$ (product rule) - $\log(a^n) = n\log a$ (power rule) First, apply the quotient rule: $\log\left(\frac{x^3y^2}{z^4}\right) = \log(x^3y^2) - \log(z^4)$ Next, use the product rule on the first term: $\log(x^3y^2) = \log(x^3) + \log(y^2)$ Finally, apply the power rule to each term: $\log(x^3) = 3\log x$, $\log(y^2) = 2\log y$, and $\log(z^4) = 4\log z$ Combining everything: $3\log x + 2\log y - 4\log z$, which is choice C. Choice A incorrectly brings the exponents inside the logarithms as coefficients of the variables. Choice B mistakenly treats the division as a fraction of logarithms rather than a difference. Choice D fails to properly separate all terms and incorrectly handles the exponents in both the numerator and denominator. Remember: exponents become coefficients when you break down logarithms, multiplication becomes addition, and division becomes subtraction. Always work systematically through the logarithm properties rather than trying to manipulate the expression intuitively.
8. Which expression represents the complete simplification of $\log_2(8x^3) + \log_2\left(\frac{2}{x}\right) - \log_2(4x^2)$?
   Answer: $\log_2(4) = 2$ Combining using logarithm properties: $\log_2(8x^3) + \log_2\left(\frac{2}{x}\right) - \log_2(4x^2) = \log_2\left(\frac{8x^3 \cdot \frac{2}{x}}{4x^2}\right) = \log_2\left(\frac{16x^2}{4x^2}\right) = \log_2(4) = 2$.

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