To prove: ln(uv) = lnu + lnv

e^ae^b = e^a+b implies

ln(e^ae^b) = ln(e^a+b)

ln(e^ae^b) = a + b (1)

Set a = ln u, b = ln v

Then (1) becomes

ln(e^lnue^lnv) = ln u + ln v

ln(uv) = lnu + lnv