Your skepticism is excessively cautious 😁. You can work around precision limits perfectly fine as long as you are aware they exist there. Multiplying your epsilon and then dividing later is a legitimate strategy, since every function is linear on a small enough scale! You can even declare that ln(1+x) ~= x and skip the logarithm calculation entirely. Using some random full precision calculator I get:
<span style="color:#323232;">ln(1+x) ~= x
</span><span style="color:#323232;">6.7/74e15 = 9.0540540...e-17
</span>
You are worried about differences in the final answer of less than 1 part in a million! I try to do my example calculations in 3 significant figures, so that’s not even a blip in the intermediate roundoffs.