Question

if 3^a = 4^b = 12^(-c) then ab+bc+ca=

Answer 1

3^a = 4^b = 12^(-c) = k (let)
3^a = k
taking log both sides,
alog3 = logk
a = logk/log3 ........(1)

similarly, 4^b = k
taking log both sides,
blog4 = logk
b = logk/log4 ...............(2)

and 12^(-c) = k
taking log both sides,
-clog12 = logk
-clog(3 × 4) = logk
-c{log3 + log4} = logk

from equations (1) and (2),
=> -c{logk/a + logk/b} = logk
=> -c{1/a + 1/b} = 1
=> -c{b + a} = ab
=> -cb - ac = ab
=> ab + bc + ca = 0

Answer 2

HELLO DEAR,

GIVEN:-

3^a = 4^b = 12^(-c) = k(say)

3^a = k

taking log both sides,

alog3 = logk

a = logk/log3

log3 = logk/a  ---------( 1 )

similarly, 4^b = k

taking log both sides,

blog4 = logk

b = logk/log4

log4 = logk/b -------------( 2 )

AND

12^(-c) = k

taking log both sides,

-clog12 = logk

-clog(3 × 4) = logk

-c{log3 + log4} = logk

from --------- (1) &-------- (2)

= -c{logk/a + logk/b} = logk

= -c{1/a + 1/b} logk = logk

= -c{(a + b/ab)} = 1

= -c{b + a} = ab

= -cb - ac = ab

=  ab + bc + ca = 0

I HOPE ITS HELP YOU DEAR,

THANKS