if a^x=b^y=c^z and b^2=ac Prove that, 1/x+1/z=2/y.
One way to prove 1/x + 1/z = 2/y with the given information is to use logarithms. By taking the logarithm in base 10, if f = k^r, then
log f = log k^r. The power (r) is moved to the front of the log, so:
log f = r log k.
Using the previous thought about logarithms with the given information.
a^x = b^y = c^z
log a^x = log b^y = log c^z
x log a = y log b = z log c (Equation 1)
Starting with 1/x + 1/z, since x = (z log c)/ log a, then
1/x = log a/(z log c) and 1/z = log c/(x log a)
1/x + 1/z = log a/(z log c) + log c/(x log a)
Using equation 1, x log a = z log c. The denominators are equal.
= log a/(z log c) + log c/(z log c)
= `(log a + log c)/(z log c)`
`=(log a*c)/(zlogc)` Using the rule that log a*c = log a + log c
`=(log b^2)/(y log b)` a*c = b^2 and z log c = y log b
The proof uses the following logarithm rules
log x^y = y log x
log x*y = log x + log y