Question

if 2^a=3^b=12^c,show that 1/c=1/b+2/a​

Answer 1

Given,

2^a = 3^b = 12^c.

To Prove,

1/c = 1/b + 2/a

Solution,

2^a = 3^b = 12^c

Let x = 2^a = 3^b = 12^c

Place log both sides in the equation.

log x = a log 2 = b log 3 = c log 12.

Calculate a value,

log x = a log 2

=> log x/log 2 = a

Calculate b value,

log x = b log 3

=> log x/log 3 = b

Calculate c value,

log x = c log 12

=> logx/log 12 = c

Then,

LHS

(1/c)

=> (1/log x/log 12)

=> (log 12/log x)

(1/b) + (2/a)

=> (1/log x/log 3) + (2/log x/log 2)

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

=> log 3 + 2 log 2/log x

=> log 12/log x

Hence, LHS = RHS

#Hope my answer helped you