Question

if a^x=b^y=c^z and a/b=c/b, prove that 2z/x+z=y/x

Answer 1

Take 
ax=by=cz=kax=by=cz=k (say).

Then , a=k1/x, b=k1/ya=k1/x, b=k1/y and c=k1/zc=k1/z

Also, b2=acb2=ac.

So, k2/y=k1/x+1/zk2/y=k1/x+1/z.

Equating the above two exponents, we have, 2/y=1/x+1/z2/y=1/x+1/z

i.e. 2/y =( z+x)/zx

or, 2z/(x+z) = y/x.

Ans. Y/X

The alternate method is awaiting don't upset it... :)

a^x=b^y=c^z=k (say)

Taking log both sides

xlog a =y log b= z log c= log k

Log a = log k/ x

Log b= log k / y

log c= log k / z

b^2= 2ac

Taking log

2 log b =log a+ log c

Subsituting the values and solving it

ANS :- Y/X


Hope this would help and please read carefully it might look a little messy ( maths equations always are.... :) :)