Question

if a^2 = b^3 = c^4 = d^5 then prove that loga (b is subset of d) = 47/30

Answer 1

Let a²=b³=c⁴=d⁵=p⁶⁰, where p is an abitrary number p≠1.

So a²=p⁶⁰, a=p³⁰; b³=p⁶⁰, b=p²⁰; c⁴=p⁶⁰, c=p¹⁵; d⁵=p⁶⁰, d=p¹².

These can be written, log(a)=30log(p), log(b)=20log(p), log(c)=15log(p), log(d)=12log(p).

So we have each number defined in terms of p. Each number can be regarded as a unique set of p values, the product of the elements of the set being the value of the number. The superset consists of 77 p’s, 30 of which belong to set a and the remaining 47 to be shared by sets b, c and d. The ratio of sets b, c and d to a is therefore 47:30 or 47/30.