# function determination

Given that a function is defined with the following properties:-

f(xy) =  f(x) + f(y)

it can be seen that this is the case for f(x) = log x

Is there any way that f(x) can be deduced analytically rather than intuitively ?
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x

Commented:
f(xy) =  f(x) + f(y)

f(x*1) = f(x) +      f(1)
f(1)=0
f(1) = f(x * 1/x) = f(x) + f(1/x)
f(x) = -f(1/x)

f(x+epsilon)-f(x)
=
f(x+epsilon)+f(1/x)
=
f((x+epsilon)/x)
=
f(1+epsilon/x)

If we assume that f is analytic:
f'(1)=lim(epsilon->0:f(1+epsilon)/epsilon)

f'(x)=
lim(epsilon->0:(f(x+epsilon)-f(x))/epsilon)
=
lim(epsilon->0:f(1+epsilon/x)/epsilon)
=
lim(epsilon->0:( f(1)+(epsilon/x)*f'(1))/epsilon)
=
f'(1)/x

so f(x) = integral(1..x:f'(1)/x)
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Commented:
Yeah. Just work it out.
log(xy)=log(x) + log(y) and manipulate it until the left and right match, or can't
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Commented:
You can determine a few things analytically:

f(x*1)  = f(x) + (f(1)      ==>  f(1) = 0

You can pick any value of x>1 as the base of the log function.
You don't have to pick 10 or e.
Once you have a base, you can calculate lots of values.

f(x) =1      ==>  the base

f(x*x)  =  f(x) + f(x) = 2   ==>   f(x^n) = n

f(sqrt(x)) =  1/2

Once you have lots of values, you can find any value by interpolation.
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Commented:
> you can find any value by interpolation.
assuming that f is analytic
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Author Commented:
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Commented:
Definition of logarithm:

Therefore:

And CQFD:

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Commented:
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