Question

If 2^x=3^y=12^z, prove that x = 2yz/y-z

Answer 1

Heya!!

Given Question is

2^x = 3^y = 12^z

let 2^x = 3^y = 12^z = k

2^x = k , 3^y = k And 12^z = k

2 = k^(1/x) , 3 = k^(1/y) And 12 = k^(1/z)

12 = k^(1/z)

2² × 3 = k^(1/z)

k^(1/2x) × k^(1/y) = k^(1/z)

k^{(1/2x) + (1/y)} = k^(1/z)

Now, Compare powers of k we have!!)

(1/2x) + (1/y) = (1/z)

1/2x = (1/z) - (1/y)

1/2x = (y - z)/zy

2x = zy / (y - z)

x = 2yz/ (y - z): Hence, proved.

Note:-

IF aⁿ = b

Then a = b^(1/n)