I know how to prove that. But I have a question about variable substitution.
Problem statement:
Prove directly from the definition $ L(x) = \int_{1}^x \frac 1tdt $ that $L\left(\frac 1a\right) = -L(a)$, by making a change of variables in the definite integral.
Proof:
$$L\left(\frac 1a\right) = \int_{1}^{\frac 1a} \frac 1tdt \tag1$$
Substitute: $$\frac 1t = u \to t = \frac 1u\tag2$$ and $$dt = \frac {-1}{u^2}\tag{2'}$$
$$L\left(\frac 1a\right) = \int_{1}^{\frac 1a} \frac 1tdt=\int_{1}^{a} -\frac {\frac 1u^2}{\frac 1u}du =-\int_{1}^{a} \frac {1}{u}du \tag3$$
$$-\int_{1}^{a} \frac {1}{u}du = -\ln(a) - \ln(1) = -\ln(a) = -L(a)\tag4$$
The question is: Why do we change $\frac 1a \to a$ in the third step?
Thank you.