Soooo many incorrect answers.
- "Overall, the argument x of ln(x) must be unitless" is incorrect. x can have any units. ln(x) is the definite integral from 1 to x of (1/x)dx. There is nothing in calculus that says the argument is unitless. The returned value is unitless only becuase of the defintion of the ln function. It is the area under of the curve of the plot of 1/x vs x. The x-axis can have units. The y-axis has units that are always the reciprocal of the x-axis because that is how the y-axis is defined. The units of the integral (the units of the area under the curve) is the product of the x-axis unit times the y-axis unit, which is always unitless 1. The result of ln(x) is unitless but x does not have to be. Other integrals do not have this property since other integrals don't have y=1/x.
For all integrals the unit of the integral is the product of the x-axis and y-axis units. And, the units always work. For example, if you integrate a plot of velocity versus time the unit of the integral is the product of the axis units; (velocity)(time)=(length/time)(time)=length. So the "area" under the curve has units of length which is correct. That is not an accident, or a special case, or magic, it is basic math.
Calculus would be useless if it didn't handle units correctly.
The Taylor series answer is also incorrect.
A Taylor series is an approximation of an integral. Ln(x) is not equal to the Taylor series (or the Maclauren series either)
The Taylor series shown is correct only for a very limited range of x. If you look at the full Taylor series for all x>0 you see that all the units cancel and the Taylor series returns a unitless number even if x has units. As it must since that is what ln(x) returns even if x has units. For the full Taylor series of ln(x) with x>0 see the second equation in https://www.efunda.com/math/taylor_series/logarithmic.cfm .
"Since logarithm is the inverse of the exponent, it MUST work for units also." Nope. When you take the log of a number (base e, base 10, base 43.538, base ...) the units cancel and the log is a unitless quantity. If you then use a power function you can not recover the units. If I tell you the log of the distance from the earth to the moon is 11.23 you can not recover the actual distance to the moon. You can use e^x to get the numeric value but the unit is lost forever. You must have additional information to recover the unit.
Calculus is 300 years old. Mathematicians, physics, chemists, engineers etc have been using it for that long. There are no "loop holes" or "secret things that only the Illuminati know" in math. If you have a formula and the units don't work the formula is wrong. The math isn't wrong, your formula is.
Unfortunately, math is generally taught without any consideration for units. The math must work with units but often in formulas it isn't clear what parts have units but they aren't shown, and what parts are unitless. The full Taylor series for ln(x) is a good example. There are a whole bunch of "1"s and there is one "2" and lots of x's. If you look closely at the series and take into account the definition of ln(x) you discover that the "1"s actually have the same units as the x's, whatever that may be. The "2" in the formula is actually unitless.
For trig functions the situation with units is slightly different. By definition trig functions take a ratio as an argument and the unit of both the numerator and denominator are the same. Therefore the argument to a trig function is unitless, though the parts of the ratio absolutely, positively, no question about it, have units. For example, for the sin function the definition of the sin of an angle is the ratio of the y value divided by the length of the hypotenuse. If you are on the unit circle the divisor is 1 and often omitted but it is still there. The units of both the y value and the hypotenuse are the same (e.g., inches, light years, nanometers, ...). So for trig functions the argument always collapses down to a unitless number. That doesn't mean the measurements (like the length of the hypotenuse) don't have units, it means the units always cancel out. If they don't, you've done something wrong.