When you are trying to find the difference of a qualitative and quantitative observation, think of this QUALITative and QUANTITative quality and quantity, qualitative is either an estimation or like the color of something. qualitative is an exact number, as in it can be measured!!
Qualitative data or whatever it is is non numerical, so it has nothing to do with numbers, while quantitative is with numbers. Think of it like this: Quantitative has part of the word quantity in it, and qualitative has part of the word quality in it. So that tells you the quantitative would be a quantity of numbers, and qualitative would be the quality of something.
![\bf ln\left( \sqrt[7]{e^4} \right)\implies log_e\left( e^{\frac{4}{7}} \right)\\\\ -----------------------------\\\\ log_{{ a}}{{ a}}^x\implies x=x\impliedby \textit{let's use this cancellation rule}\\\\ -----------------------------\\\\ log_e\left( e^{\frac{4}{7}} \right)\iff \cfrac{4}{7}](https://tex.z-dn.net/?f=%5Cbf%20ln%5Cleft%28%20%5Csqrt%5B7%5D%7Be%5E4%7D%20%5Cright%29%5Cimplies%20log_e%5Cleft%28%20e%5E%7B%5Cfrac%7B4%7D%7B7%7D%7D%20%5Cright%29%5C%5C%5C%5C%0A-----------------------------%5C%5C%5C%5C%0Alog_%7B%7B%20%20a%7D%7D%7B%7B%20%20a%7D%7D%5Ex%5Cimplies%20x%3Dx%5Cimpliedby%20%5Ctextit%7Blet%27s%20use%20this%20cancellation%20rule%7D%5C%5C%5C%5C%0A-----------------------------%5C%5C%5C%5C%0Alog_e%5Cleft%28%20e%5E%7B%5Cfrac%7B4%7D%7B7%7D%7D%20%5Cright%29%5Ciff%20%5Ccfrac%7B4%7D%7B7%7D)