What are all solution of the equation 289^x-12*17^x+11=0 ?

Since 289 is a power of 17, we'll
write:

289 = 17^2

We'll raise
both sides by x;

289^x =
17^2x

We'll re-write the
equation:

17^2x - 12*17^x + 11 =
0

We'll replace 17^x by 
t:

t^2 - 12t + 11 = 0

t^2 - t
- 11t + 11 = 0

t(t - 1) - 11(t-1) =
0

(t - 1)(t - 11) = 0

We'll
cancel each factor and w'ell get:

t - 1 = 0 => t1 =
1

t - 11 = 0 => t2 =
11

17^x = t1 <=> 17^x  = 1 <=>
17^x = 17^0

Since the bases are matching, we'll apply one
to one property:

x = 0

17^x =
t2<=> 17^x  = 11 <=> ln 17^x = ln 11 => x = ln 11/ln
17

The complete set of solutions of the
equation is {0 ; ln 11/ln 17}.