Question: P13 - ENGAA 2021 Q29
What is the mean of
log10 27, log10 64, log10 216?
A log10 307
3
B log10 81
3
C log10 (6^12)
3
D log10 64
E log10 72
F log10 108
2022
P14 - NSAA 2022 Q1 / ENGAA 2022 Q1
Which one of the following is a simplification of
y ( (3x^1/2 z) / y^3 )^2 ?
1. **Problem 1: Find the mean of $\log_{10} 27$, $\log_{10} 64$, and $\log_{10} 216$.**
2. The mean of logarithms is the average: $$\text{mean} = \frac{\log_{10} 27 + \log_{10} 64 + \log_{10} 216}{3}$$
3. Use the logarithm property: $$\log a + \log b + \log c = \log (a \times b \times c)$$
4. So, $$\text{mean} = \frac{\log_{10} (27 \times 64 \times 216)}{3}$$
5. Calculate the product inside the logarithm:
$$27 = 3^3$$
$$64 = 2^6$$
$$216 = 6^3 = (2 \times 3)^3 = 2^3 \times 3^3$$
6. Multiply:
$$27 \times 64 \times 216 = 3^3 \times 2^6 \times (2^3 \times 3^3) = 3^{3+3} \times 2^{6+3} = 3^6 \times 2^9$$
7. So,
$$\text{mean} = \frac{\log_{10} (3^6 \times 2^9)}{3} = \frac{\log_{10} (3^6) + \log_{10} (2^9)}{3} = \frac{6 \log_{10} 3 + 9 \log_{10} 2}{3}$$
8. Simplify the fraction:
$$= 2 \log_{10} 3 + 3 \log_{10} 2$$
9. Rewrite as a single logarithm:
$$= \log_{10} (3^2) + \log_{10} (2^3) = \log_{10} (9) + \log_{10} (8) = \log_{10} (9 \times 8) = \log_{10} 72$$
10. **Answer for Problem 1:** $\boxed{\log_{10} 72}$ which corresponds to option E.
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11. **Problem 2: Simplify $y \left( \frac{3x^{1/2} z}{y^3} \right)^2$.**
12. Apply the exponent to the fraction inside the parentheses:
$$y \times \frac{(3)^2 (x^{1/2})^2 (z)^2}{(y^3)^2} = y \times \frac{9 x^{1} z^{2}}{y^{6}}$$
13. Simplify powers:
$$= y \times \frac{9 x z^{2}}{y^{6}} = \frac{9 x z^{2} y}{y^{6}}$$
14. Cancel common factors of $y$:
$$= 9 x z^{2} \times \frac{\cancel{y}}{y^{6}} = 9 x z^{2} y^{1-6} = 9 x z^{2} y^{-5}$$
15. Rewrite with positive exponents:
$$= \frac{9 x z^{2}}{y^{5}}$$
16. **Answer for Problem 2:** $\boxed{\frac{9 x z^{2}}{y^{5}}}$