Solve the equation log3 x + 1/log3 x=5/2

We'll multiply all terms both sides by the least common
denominator.

2*log3 (x)*log3 (x) + 2 = 5*log3
(x)

2*[log3 (x)]^2 - 5*log3 (x) + 2 =
0

We'll replace log3 (x) by
t.

2t^2 - 5t + 2 = 0

We'll
apply quadratic formula:

t1 = [5+sqrt(25 -
16)]/4

t1 = (5+3)/4

t1 =
2

t2 = 1/2

But log3
(x)=t.

log3 (x)=t1 <=> log3
(x)=2

We'll take antilogarithms and we'll
get:

x = 3^2

x =
9

log3 (x)=t2

x = sqrt
3

Since both values are positive, we'll
accept them as solutions of equation: {sqrt3 ;
9}.