Find all values of 'n' in this equation: 4^n + 4^(-n) = 14

I'm sure, that n isn't natural number, because:

4^1 + 4^(-1) = 4 + (1/4)^1 = 4 + 1/4 = 4.25

and

4^2 + 4^(-2) = 16 + (1/4)^2 = 16 + 1/16 = 16.0625

14 is greater than 4.25 and is lower than 16.0625, so:

1 < n < 2

I don't have any idea what's next. Can you help me?

x + 1/x = 14 or

x² - 14x + 1 = 0

which is just a quadratic equation in x. Once you found it,

n = logx / log4 = log₄x = (log₂x)/2.

4^n + 4^(-n) = 14

(4^n)^2 + 1 = 14*4^n

(4^n)^2 - 14*4^n + 1 = 0

so using the quadratic formula -

4^n = (14 +/- (14^2 - 4)^0.5) / 2

4^n = 7 +/- 48^0.5

then by logging both sides

n log 4 = log (7 +/- 48^0.5)

n = log (7 +/- 48^0.5) / log 4

n = 1.89997 , -1.89997