If log₁₀ 2 = a and log₁₀ 3 = b, express log₅ 12 in terms of a and b — JEE Mathematics
If $\log_{10} 2 = a$ and $\log_{10} 3 = b$, express $\log_{5} 12$ in terms of $a$ and $b$.
1 Answer
MPMeera Pillai
✓ Accepted
· 2mo ago
▲ 36
Using the change of base formula, express $\log_5 12$ with base 10:
$$\log_5 12 = \frac{\log_{10} 12}{\log_{10} 5}$$
Simplify the numerator:
$$\log_{10} 12 = \log_{10}(2^2 \times 3) = 2\log_{10} 2 + \log_{10} 3 = 2a + b$$
Simplify the denominator:
$$\log_{10} 5 = \log_{10}\left(\frac{10}{2}\right) = \log_{10} 10 - \log_{10} 2 = 1 - a$$
Combine them back:
$$\log_5 12 = \frac{2a + b}{1 - a}$$
Answer: $\frac{2a + b}{1 - a}$
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Discussion (4)
P
For revision — the key formula used here comes up almost every year.
A
This finally made it click for me — thank you!
S
This is exactly the kind of step-by-step I needed. Respect.
AR
What changes if the medium/conditions were different?