Why is -ln x is not equal to 1/ln x?

Question

I am doing differential equation now and I need to convert them into the proper form in order to do my homogeneous differential equation. So now I just found out that -ln x is not equal to 1 / ln x. I thought it should be able to convert to ln x to the negative 1 then I can put it into the form 1/ ln x. Can anyone explain about it?

Answer 1

Why is $-\ln(x)$ is not equal to $\frac{1}{\ln(x)}$ ?

Because $\space -\ln(x)=\ln(\frac{1}{x})\space $ and $\space \ln(\frac{1}{x})\space $ is not equal to $\space \frac{1}{\ln(x)}$

In general, for most of the functions $f(x)$ we don't have $f(\frac{1}{x})=\frac{1}{f(x)}$