CAT 2021 Slot 3 · Quant · Question 6
CAT 2021 Slot 3 previous year question on Logarithms, with the answer and a step-by-step solution.
For a real number a, if \frac{{\log}_{15}a+{\log}_{32}a}{\left({\log}_{15}a \right)\left({\log}_{32}a \right)}=4 then a must lie in the range
- A4<a<5
- B3<a<4
- Ca>5
- D2<a<3
Pick an option, then check.
Answer & solution
Answer: Option A
Split the fraction rather than expanding it:
\frac{{\log}_{15}a+{\log}_{32}a}{\left({\log}_{15}a \right)\left({\log}_{32}a \right)}=\frac{1}{{\log}_{32}a}+\frac{1}{{\log}_{15}a}.
By the reciprocal rule \frac{1}{{\log}_{b}a}={\log}_{a}b, this is {\log}_{a}32+{\log}_{a}15={\log}_{a}\left( 32\times 15 \right)={\log}_{a}480.
So {\log}_{a}480=4, that is {a}^{4}=480.
Since {4}^{4}=256 and {5}^{4}=625, and 256<480<625, we get 4<a<5.
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