How to simplify the expression $\frac{\ln(t)}{\ln(t')}?$

Question

I am solving a Cauchy problem: $x'+\frac{x}{t\ln t}=\frac{1}{\ln t'}, \ \ x(e)=e$ and I came to an expression $\frac{\ln t}{\ln t'},$ that I don't know how to simplify or express differently. Can someone help?

Answer 1

There is something wrong in the wording of your question because $x(t)=\frac{t}{\ln(t')}$ is not solution of the equation $$x'+\frac{x}{t\ln (t)}=\frac{1}{\ln (t')}$$ $x(t)=\frac{t}{\ln(t')}$ is solution of the equation $$x'+\frac{x}{t\ln (t)}=\frac{1}{\ln (t)}$$ Possibly there is a confusion between $\quad x'+\frac{x}{t\ln t}=\frac{1}{\ln t},\:...\quad$ and $\quad x'+\frac{x}{t\ln t}=\frac{1}{\ln t'}…\quad$ that is a confusion between virgule and apostrophe, due to unclear typography.

Also, forgetting the parentheses can lead to confusion.