And the derivative of ln(2x) is also 1/x, and same for ln(3x), etc.
So, since they all have the same derivative then either they are the same line, or parallel lines.
Graphing indicates that they are parallel.
The difference between the y values of f(x)=ln(x) and f(x)=ln(2x) for any x is 0.693; between ln(2x) and ln(3x) it’s 0.405; then 0.288, 2.223, and so forth.
So, my question, where does the difference is values originate from -> if the rate of change at any given x is identical between the two functions, where/when/how did the lines end up in different places?
Most parallel lines involve the addition of a constant after manipulating the x term, but that isn’t the case here. So what’s going on?
eta: could a mod fix the typo in the thread title?
Logarithms turn multiplication into addition. E.g., ln(a * b) = ln(a) + ln(b).
In particular, ln(2x) = ln(x) + ln(2). ln(3x) = ln(x) + ln(3). And so on. So it is just addition of a constant. Any two functions with the same derivative have a constant difference, as you yourself pointed out and even calculated, yet seemed to fail to fully appreciate.
Sometimes when I have issues thinking about ln(), I rethink it as log10() to work through the logic. Using log base10 is my mental crutch because my brain can’t calculate natural logs (2.71828 ^ ??? = ???) with arbitrary concrete numbers.