JEE Main Maths: Logarithms and Their Properties questions with solutions

Q1. Let a, b be positive real numbers satisfying log₉(a) = log₁₅(b) = log₂₅(2a + b). Find the value of b/a.

1. 1
2. 2
3. 3
4. 4

Answer: 2

Setting log₉(a) = log₁₅(b) = log₂₅(2a+b) = k gives a=9^k, b=15^k, 2a+b=25^k. Substituting: 2*9^k + 15^k = 25^k. Let t = (5/3)^k; dividing by 9^k gives 2 + t = t², so t² - t - 2 = 0, giving t = 2 (positive). Then b/a = 15^k/9^k = (5/3)^k = t = 2.