Help with a couple of things

Hey there, mmm, you will see, I'm a calculus student, today as I was coming back home I remembered the logarythm properties, the thing is, I found out I just accepted them as they told me, and never questioned them, so as soon as I got home I tried to analyze the properties from a deeper perspective, I was wondering, there's someone that can explain me and demonstrate me the reason of the properties in logarythms? I would really appreciate it

Also, as I was experimenting I came with a log 9, but I (feeling guilty of not having questioned the properties told to me before) decided to solve them using differential calculus instead of a calculator so I made like this:

f(x) = log x

f'(x) = 1/(x · ln 10)

f(9) = f'(10)·dx + f(10)

log 9 = -1/(10·ln 10) + 1

So I found another logarythm, ln 10

g(x) = ln x

g'(x) = 1/x

g(e + (10-e)) = g'(e)·dx + g(e)

ln 10 = (10-e)/e + 1

So... I decided to solve them, mmm... the second one is a total fail, real approximation is 2.303, my approx is 3.679

I also tested the first one as its (using the calculator to solve the ln 10) and it gave me 0.9565, while the real approx is 0.9542 (not so much difference 0,002)

My question is, what am I doing wrong?

The reason of why my second differential failed is because the differential (10-e) is to big?

Thanks for your time by the way