Given ln3/ln12 = x and ln5/ln12 = y calculate[log15(400)]/2 .

We'll re-write the numbers x and
y:

ln 3/ln 12  = log 12 (3) = x and ln 5/ln 12 = log 12 (5)
= y

We also re-write the number that has to be
calculated:

[log 15 (400)]/2 = log 15 (sqrt 400) = log 15
(20)

We'll add log 12 (3)=x and log 12 (5)=y, we'll
get:

log 12 (3) + log 12 (5) = x + y
(1)

Since the bases are matching, we'll apply the product
rule:

log 12 (3) + log 12 (5) = log 12
(3*5)

log 12 (3) + log 12 (5) = log 12 (15)
(2)

We'll substitute (1) in
(2):

x + y = log 12 (15)

But
log 12 (15) = 1/log 15 (12)

1/log 15 (12) = x +
y

log 15 (12) = 1/(x + y)
(3)

log 15 (12) = log 15
(4*3)

log 15 (4*3) = log 15 (4) + log 15
(3)

log 15 (4) = log 15 (12) - log 15 (3)
(*)

Now, we'll
calculate log 15 (20):

log 15 (20) = log 15
(4*5)

log 15 (4*5) = log 15 (4) + log 15
(5)

log 15 (4) = log 15 (20) - log 15 (5)
(**)

We'll write log 15 (3) with respect to
log 12 (3):

log 15 (3) = log 12 (3)*log 15
(12)

log 15 (3) =
a*1/(x+y)

We'll write log 15 (5) with respect to log 12
(5):

log 15 (5) = log 12 (5)*log 15
(12)

log 15 (5) =
y*1/(x+y)

Equating (*) = (**), we'll
have:

log 15 (12) - log 15 (3) = log 15 (20) - log 15
(5)

We'll add log 15 (5) both
sides:

log 15 (20) = log 15 (12) - log 15 (3) + log 15
(5)

log 15 (20) =
(1+y-x)/(x+y)