If x=log3/5, y=log5/7, z=log root7/3, find the value of x+y+z

Given:

x=log (3/5)

y=log (5/7)

z=log (sqrt (7/3)

Find x+y+z

Steps:

Using the law of logarithms, log A + log B = log (AB), we get

x+y+z

=log (3/5) + log (5/7) + log (sqrt (7/3)

=log (3/5*5/7*sqrt (7/3))

=log (3/7*sqrt (7/3)

=log ((sqrt (3/7) * sqrt (3/7) * sqrt (7/3))

=log ((sqrt (3/7) * sqrt (3/7*7/3))

=log (sqrt (3/7) * 1)

=log (sqrt (3/7))

Now apply another law of logarithms, log (sqrt (A)) = (1/2) log (A)

= (1/2) log (3/7)

which is the final answer.