Do any positive real numbers alpha and beta exist such that lnalpha times beta equals lnalpha plus lnbeta and if so what are they?
I assume the question is NOT about ln(a*b) = ln(a) + ln(b)
because that is true for all positive real a and b.
Instead, you want a solution to
ln(a) * b = ln(a) + ln(b)
or, ln(a) * (b-1) = ln(b)
ln(a) = ln(b)/(b-1)
ln(a) = ln[b1/(b-1)]
Exponentiating, a = b1/(b-1)
For any real number b > 1, a given by the above equation will
meet your requirements.