Lots of ideas however mostly not working, for example:
Assume there would be a function \( f \) with
\( f(x^y)=f(x)f(y) \)
Then surely
\( f(x)f(x^n)=f(x^{x^n})=f((...((x^x)^x)\dots)^x)=f(x)^{n+1} \)
So if there is an \( x \) with \( f(x)\neq 0 \) we have
\( f(x^n)=f(x)^n \).
for \( y=x^n \) we also have:
\( f(y^{1/n})=f(x)=f(y)^{1/n} \)
and both together we get
\( f(x^{m/n})=f(x)^{m/n} \)
If we assume that \( f \) is continuous then this is valid not only for rationals but also for reals \( \alpha \):
\( f(x^\alpha)=f(x)^\alpha \).
No let \( x=c \) constant and let \( \alpha \) be variable:
\( f(c^t)=f( c)^t \)
\( f(x)=f(c^{\log_c(x)})=f( c)^{\log_c(x)}=x^{\log_c(f( c))}=x^d \)
So \( f(x)=x^d \) for some constant \( d \). But then:
\( (x^y)^d=f(x^y)=f(x)f(y)=x^d y^d \)
if \( d\neq 0 \):
\( x^y=xy \)
which can not be satisfied for all \( x \), \( y \).
Hence either \( d=0 \) or \( f(x)=0 \) which we excluded in our previous considerations.
Proposition. Let \( f \) be a continuous function defined on the positive reals such that \( f(x^y)=f(x)f(y) \) for all \( x,y \), then either \( f(x)=0 \) for all \( x \) or \( f(x)=1 \) for all \( x \).
Ivars Wrote:3_log(a^b) =3_log(a[3]b)= 3_log(a)*3_log(b) = 3_log(a)[2]3_log(b)
Assume there would be a function \( f \) with
\( f(x^y)=f(x)f(y) \)
Then surely
\( f(x)f(x^n)=f(x^{x^n})=f((...((x^x)^x)\dots)^x)=f(x)^{n+1} \)
So if there is an \( x \) with \( f(x)\neq 0 \) we have
\( f(x^n)=f(x)^n \).
for \( y=x^n \) we also have:
\( f(y^{1/n})=f(x)=f(y)^{1/n} \)
and both together we get
\( f(x^{m/n})=f(x)^{m/n} \)
If we assume that \( f \) is continuous then this is valid not only for rationals but also for reals \( \alpha \):
\( f(x^\alpha)=f(x)^\alpha \).
No let \( x=c \) constant and let \( \alpha \) be variable:
\( f(c^t)=f( c)^t \)
\( f(x)=f(c^{\log_c(x)})=f( c)^{\log_c(x)}=x^{\log_c(f( c))}=x^d \)
So \( f(x)=x^d \) for some constant \( d \). But then:
\( (x^y)^d=f(x^y)=f(x)f(y)=x^d y^d \)
if \( d\neq 0 \):
\( x^y=xy \)
which can not be satisfied for all \( x \), \( y \).
Hence either \( d=0 \) or \( f(x)=0 \) which we excluded in our previous considerations.
Proposition. Let \( f \) be a continuous function defined on the positive reals such that \( f(x^y)=f(x)f(y) \) for all \( x,y \), then either \( f(x)=0 \) for all \( x \) or \( f(x)=1 \) for all \( x \).