Basic Maths and Logarithm
Properties of Logarithms
Premium Question
Grade 11
Question:
If $\log_3 2 = a$ and $\log_5 3 = b$, then $\log_{15} 20$ equals:
(1) $\frac{2ab + 1}{b + 1}$
(2) $\frac{2ab + 1}{a + 1}$
(3) $\frac{ab + 2}{b + 1}$
(4) $\frac{ab + 1}{2b + 1}$
Step-by-Step Solution
Key Concept: Let $\log 5 = s, \log 3 = m, \log 2 = l$. Then $l/m = a, m/s = b$, so $l = abs$. $\log_{15} 20 = \log 20 / \log 15 = (2l + s)/(m + s) = (2abs + s)/(bs \cdot s^{-1} \cdot s + s) \dots$ work in base-10 with $l = abs, m = bs$: $(2abs + s)/(bs + s) = s(2ab + 1)/s(b + 1) = (2ab + 1)/(b + 1)$.
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Correct Answer: (1)