Math Olympiad l A Nice Radical Exponential Equation l VIJAY Maths

[@ganeshdas3174]

On squaring B/S
4^x - 2 6^x + 9^x = 6^x - 9^x
4^x - 3.6^x + 2 .9^x = 0 devide it by 4^x B/S
1 - 3(3/2) ^x + 2.(3/2) ^2 x = 0
Put (3/2) ^x = p
2 p^2 - 3 p + 1= 0
Solve the quadratic eqn.
p =( 3+ -√9 - 8 ) /4
p = 1& 1/2
Recall p= (3/2) ^x = 1 , x = 0
(3/2) ^x = 1/2, take log
x = log( 1/2)/(log 3/2) & 0

[@adribber]

2ⁿ - 3ⁿ = √(6ⁿ - 9ⁿ)
2ⁿ - 3ⁿ = √{3ⁿ(2ⁿ - 3ⁿ)}
(2ⁿ - 3ⁿ)² = [√{3ⁿ(2ⁿ - 3ⁿ)}]²
(2ⁿ - 3ⁿ)² = 3ⁿ(2ⁿ - 3ⁿ)
(2ⁿ - 3ⁿ)² - 3ⁿ(2ⁿ - 3ⁿ) = 0
(2ⁿ - 3ⁿ){(2ⁿ - 3ⁿ) - 3ⁿ} = 0
(2ⁿ - 3ⁿ)(2ⁿ - 2*3ⁿ) = 0
2ⁿ - 3ⁿ = 0 , 2ⁿ - 2*3ⁿ = 0
Case 1 : 2ⁿ - 3ⁿ = 0
2ⁿ = 3ⁿ
log2ⁿ = log3ⁿ
log2ⁿ - log3ⁿ = 0
nlog2 - nlog3 = 0
n(log2 - log3) = 0
∴ n = 0
Case 2 : 2ⁿ - 2*3ⁿ = 0
2ⁿ = 2*3ⁿ
log2ⁿ = log(2*3ⁿ)
nlog2 = log2 + nlog3
nlog2 - nlog3 = log2
n(log2 - log3) = log2
∴ n = log2/(log2 - log3)

[@RealQinnMalloryu4]

{4 ➖ 6}=2 (x ➖ 2x+2). {36 ➖ 81} =45 3^15 23^3^5 1^3^1 3^1(x ➖ 3x+1).