Do any positive real numbers alpha and beta exist such that lnalpha times beta equals lnalpha plus lnbeta and if so what are they?

Answer 1

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.