What is x if 13^x - 20=13^(3-x) ?

We'll remember the quotient property of
exponentials:

13^(3-x) =
13^3/13^x

We'll re-write the
equation:

13^x - 20 - 13^3/13^x =
0

We'll multiply by 13^x both
sides:

13^2x - 20*13^x - 13^3 =
0

We'll substitute 13^x =
t:

t^2 - 20t - 2197 = 0

We'll
apply quadratic rule:

t1 = [20+sqrt(400 +
8788)]/2

t1 = (20 +
sqrt9188)/2

t1 =
10+sqrt2297

t2 =
10-sqrt2297

13^x =
10+sqrt2297

log 13^x = ln
(10+sqrt2297)

x = log (10+sqrt2297)/ln
13

x = 1.5825 approx.

13^x =
10 - sqrt2297 impossible! ( since 10 < sqrt2297 => 13^x = negative value,
that is impossible).

The equation has only
one solution, x = 1.5825 approx.