calculus - Logarithm defined using the definite integral without Fundamental Theorem

If one wants to define the natural logarithm using (Riemann) integrals, he could do as follows:

$$ \log(x) := \int_{1}^{x} \frac{1}{t} dt $$

Let's assume we hadn't defined the $\exp$-function yet. How can one prove some basic logarithmic identities WITHOUT using the Fundamental Theorem of Calculus.

1. $\log (xy) = \log (x) + \log (y)$


2. $\log '(1) = 1$

Edit:
This exercise came up as an homework for a Calculus I class at my university. They explicitly stated not to use the exp-function or the FTC. Since I didn't know how they were supposed to solve this with these restrictions, I thought I post this problem here. (Note: This homework was due a few weeks ago. So posting the solution here shouldn't be a problem.)