Home page for accesible maths Math 101 Chapter 2: Functions of a real variable 2.34 The log function

\log

\ln

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

2.35 Properties of log

The graph of $\log:{\mathbb{R}}^{+}\rightarrow{\mathbb{R}}:$ $y=\log x$ is obtained from that of exp simply by reversing the rôles of $x$ and $y$ . Since $y=e^{x}$ implies that $x=\log y$ , we can say that $\log y$ is the power to which $e$ must be raised to give $y$ .

Properties of the natural logarithm

The domain of $\log$ is $\{x\in{\mathbb{R}}:x>0\}$ and its range is ${\mathbb{R}}.$ The log function satisfies:

(i) $\log(\exp x)=x$ for all $x\in{\mathbb{R}}$ ;

(ii) $\exp(\log x)=x$ for all $x>0;$

(iii) the functional equation of log is

\log(xy)=\log x+\log y\qquad(x,y>0);

(iv) $\log x\rightarrow\infty$ as $x\rightarrow\infty$ ; whereas $\log x\rightarrow-\infty$ as $x\rightarrow 0+.$