Functional Equations

The Cauchy-type relation f(xy) = f(x) + f(y) characterizes logarithm-like functions and is a recurring JEE Advanced functional-equation pattern. Setting y = x gives f(x^2) = 2f(x), and more generally f(x^n) = n f(x) by induction. Since 32 = 2^5, we get f(32) = f(2^5) = 5 f(2) = 5·3 = 15. The additive-to-multiplicative structure means f behaves like a constant multiple of a logarithm base 2. Option 10 uses 2^5 incorrectly as 5·2. Option 12 applies the wrong exponent of 4. Option 96 mistakenly multiplies 32 by 3 as though f were linear in its argument. Hence f(32) = 15. Plausibility check: f(4) = 2f(2) = 6 and f(8) = 3f(2) = 9 follow the same rule, and f(32) = f(8)·? no—f(8·4)=f(8)+f(4)=9+6=15, independently confirming the value through a different factorization of 32. Cauchy-type functional equations characterize logarithms, exponentials, additive or linear maps depending on which operation pairing they encode, and the multiplicative-to-additive form here is the logarithmic signature. The pivotal step is anchoring the entire family of values to a single known input, here f of two, after which induction propagates that seed to every power, making the whole structure determinate from one datum and the rule.