Question

prove that the solution of the equation 1^x+6^x+8^x=9^x is=?​

Answer 1

Step-by-step explanation:

Dividing both sides by 4x , we get

1+(64)x=(94)x

⟹1+(32)x=[(32)x]2

Let a=(32)x

⟹1+a=a2

⟹a2−a−1=0

⟹a2−2⋅12⋅a+14−54=0

⟹(a−12)2=54

⟹a−12=±5√2

⟹a=1±5√2

⟹(32)x=1±5√2

Side note: This is awkwardly clumsy. I'll express 1+5√2 as φ , 1−5√2 as φ−5–√ and 32 as 1.5 for the sake of convenience.

⟹1.5x={φφ−5–√

⟹x={log1.5φlog1.5(φ−5–√)

But, φ−5–√ is a negative number. And I'll stick to real solutions. Hence, I won't consider log1.5(φ−5–√) as a solution here. However, it is a solution and is a complex number.

∴x=log1.5φ

SG