Question

if 3^x=4^x-1 then x=​

Answer 1

4^x - 3^x = 1

There's a formula:

b^n - a^n = (b-a)((a^0)×b^(n-1) + (a^1)×b^(n-2) +..........(a^n)×b^0)

Substituting the values we get:

(1)(4^(x-1) + ............ + 3^(x) =1

Unless we know value of x, this equation is an Infinitely long sum that sums up to 1

Example: 9/10 + 9/100 + 9/1000... = 0.99999999......~1

By intuition, we can say that each term is a fraction greater than 0 but less than 1, and also x<1 (because, if x=1, the first term itself becomes 1, remaining all turn to 0, which is highly impossible!)

So the exponent must be negative.

Therefore, x ranges from 0.00000000......1>0 (An Infinitely small number) to 0.99999999.....<1

(A number so close to 1 that we consider its value to be exactly 1)

However, there will be one or a few values of x that satisfy, but we can't exactly tell the value. Atleast now.