Solve for x 6^x + 4^x = 9^x

Given

4ˣ + 6ˣ = 9ˣ

Find out

The value of x in a given equation

Solution

On dividing both sides by 4ˣ we get

=> 1 + (6/4)ˣ = (9/4)ˣ

=> 1 + (3/2)ˣ = ((3/2)²)ˣ

=> ((3/2)²)ˣ – (3/2)ˣ – 1 = 0

=> ((3/2)ˣ )² – (3/2)ˣ – 1 = 0

Let us assume (3/2)ˣ = y

=> y² – y – 1 = 0

=> y = (1 ± √5)/2

(3/2)ˣ = (1 ± √5)/2

On taking log both sides we get,

x log(3/2) = log ((1 + √5)/2 ) as log of -v2 not defined

=> x = log( (1 ± √5)/2 ) / log(3/2)

Alternative

4ˣ + 6ˣ = 9ˣ

Dividing both sides by 9ˣ

=> (4/9)ˣ + (6/9)ˣ = 1

=> ((2/3)ˣ)² + (2/3)ˣ – 1 = 0

=> (2/3)ˣ = (-1 ± √5)/2

=> x = log ((-1 + √5)/2)/log(2/3)

Answer

x = log( (1 ± √5)/2 ) / log(3/2)

x = log ((-1 + √5)/2)/log(2/3)

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