What is x if 9^x −10*3^(x−1) +1= 0?

We'll create matching bases for all the exponential
terms.

9^x = 3^2x

3^(x-1) =
3^x/3

We'll re-write the
equation:

3^2x - 10*3^x/3 + 1 =
0

We'll replace 3^x by t;

t^2
- 10t/3 + 1 = 0

3t^2 - 10t + 3 =
0

We'll apply the quadratic
formula:

t1 = [10+sqrt(100 -
36)]/6

t1 = (10+8)/6

t1 =
3

t2 = 1/3

t1 = 3^x =>
3 = 3^x

Since the bases of exponentials are matching, we'll
apply one to one rule:

x =
1

t2 = 3^x => 1/3 = 3^x => 3^-1 =
3^x

x = -1

The
soutions of the equation are {-1 ; 1}.