Solve for x log2 (x) +log3 (x)=1.

We'll change the bases of logarithms into the common base
10.

log2 (x) = log x/log
2

log3 (x) = log x/log 3

We'll
re-write the equation:

log x/log 2 + log x/log 3 =
1

We'll calculate the LCD = (log 2)*(log
3)

We'll multiply all over by (log 2)*(log
3):

(log 2)*(log 3)*(log x)/log 2 + (log 2)*(log 3)*(log
x)/log 3 = (log 2)*(log 3)

(log 3)*(log x) + (log 2)*(log
x) = (log 2)*(log 3)

We'll factorize by log
x:

(log x)*[(log 3) + (log 2)] = (log 2)*(log
3)

We'll apply the product property of
logarithms:

log x = (log 2)*(log
3)/log(2*3)

log x = (log 2)*(log 3)/log
6

Since the base is 10, we'll take antilog and we'll
get:

x = 10^[(log 2)*(log 3)/log
6]

The solution of the given equation is x =
10^[(log 2)*(log 3)/log 6].