2.36 Proofs of the properties of logs

(i) this is the definition;

(ii) this is the definition, but note that we need $x>0$ to define $\log x$.

(iii) Let $x=e^{a}$ and $y=e^{b}$ where $a=\log x$ and $b=\log y$; then

so

(iv) First let $x=e^{a}$, so $a=\log x\rightarrow\infty$ as $x\rightarrow\infty$.

Next let$x\rightarrow 0+$ so $1/x\rightarrow\infty$ hence