Properties of the Natural Logarithm: Values at 1 and e, Products, Quotients, Monotonicity, Continuity, Derivative and the Bound log t at most t - 1

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\exp:\mathbb{R}\to\mathbb{R}$ be the exponential function, let $e=\exp(1)$ be the number of The Real Exponential Function (iii), let $(0,\infty)$ be the ray of positive reals, and let $\log:(0,\infty)\to\mathbb{R}$ be the natural logarithm.

(i) $\log(1)=0$ and $\log(e)=1$.

(ii) $\log(st)=\log(s)+\log(t)$ for all $s,t\in(0,\infty)$.

(iii) $\log(1/t)=-\log(t)$ and $\log(s/t)=\log(s)-\log(t)$ for all $s,t\in(0,\infty)$.

(iv) $\log(s)<\log(t)$ for all $s,t\in\mathbb{R}$ with $0<s<t$.

(v) $\log$ is continuous on $(0,\infty)$.

(vi) For every interval $I\subseteq(0,\infty)$ containing two distinct points, the restriction $\log|_{I}:I\to\mathbb{R}$ is differentiable, and its derivative at every $t\in I$ is $(\log|_{I})'(t)=1/t$.

(vii) $\log(t)\le t-1$ for every $t\in(0,\infty)$.