Solve equation 9^x - 3^x - 1= 0?

Question

Error, I meant: 9^x-3^x-12=0

Answer 1

3^(2x) - 3^x - 1 = 0
Let y = 3^x
y ² - y - 1 = 0
y = [ 1 ± √(1 + 4) ] / 2
y = [ 1 ± √(5) ] / 2
3^x = [ 1 ± √(5) ] / 2
x log 3 = log { [ 1± √(5) ] / 2 }
x = ( log ( 1 ± √(5) ) / ( 2 log 3 )

Answer 2

9^x -3^x = 1

Rewrite as:
(3^x)² - 3^x = 1

let y = 3^x

y² - y - 1 = 0

y= (1/2) ± (1/2)sqrt(5)

Now go back to y = 3^x and take the log of both sides

ln(y) = ln(3^x) = x * ln(3)

x = ln(y)/ln(3) = ln((1/2) ± (1/2)sqrt(5)) / ln(3)

This was a challenging problem, what grade level are you in?