Question

if 2^x=3^y=6^z show that 1/x+1/y-1/z=0

Answer 1

Answer:Given 2^x=3^y=6^-z.

Lets assume that each and every term is equal to k. Which implies

2^x=k. Now by applying logarithm on both sides we get x=log k to base 2 and

1/x= log 2 to base k

And 3^y=k. Again by applying logarithm on both sides we get y=log k to base 3 and

1/y=log 3 to base k

Coming to third term we get 6^-z =k. Now by applying logarithm on both sides we get

- z=log k to base 6 which implies

z=-log k to base 6 and upon further simplification we get

Thus 1/z will be equal to - log 6 to base k.

Now upon adding 1/x+1/y+1/z will be equal to

log 2 to base k +log 3 to base k - log 6 to base k

Which will be equal to log 6 to base k - log 6 to base k. Thus makes the whole equation equal to zero.

Thus 1/x+1/y+1/z=0

Thank you :)