Logarithms

Definition: Let $a\in R_{+}^{*},\, a\ne 1$ and $b\in R_{+}^{*}$ be two real numbers. The logarithm of the strictly positive real number b is the exponent to which the number a, called the base, must be raised in order to obtain the number b.

The logarithm of b to base a is written ${{\log }_{a}}b$, $a\in R_{+}^{*},\,a\ne 1$,$b\in R_{+}^{*}$

${{\log }_{a}}b=x\Leftrightarrow {{a}^{x}}=b$ clearly $b={{a}^{{{\log }_{a}}b}}$

For $a=10$ we obtain common logarithms (lg b), and for $a=e$ we obtain natural logarithms (ln b).

${{e}^{a\ln x}}={{x}^{a}}\Rightarrow {{e}^{\ln x}}=x$

lg10=1	lne=1
lg100=lg(102)=2	ln(e2)=2
lg1000=lg(103)=3	ln(e3)=3
lg(10n)=n	ln(en)=n

Properties:

1)${{\log }_{a}}b={{\log }_{a}}c\Leftrightarrow b=c,\left( b,c>0 \right);$

2)${{\log }_{a}}a=1;$

3)${{\log }_{a}}1={{\log }_{a}}{{a}^{0}}=0;$

4)${{\log }_{a}}{{a}^{c}}=c\,;\,{{\log }_{a}}\frac{1}{b}=-{{\log }_{a}}b;\,$ ${{\log }_{a}}{{x}^{2n}}=2n{{\log }_{a}}\left| x \right|\,,x\ne 0;$

5)${{\log }_{a}}\sqrt[m]{b}=\frac{1}{m}{{\log }_{a}}b,\,\left( b>0,m\in N,m\ge 2 \right)\,;$

6)${{\log }_{a}}b\cdot {{\log }_{b}}a=1;$

7. Change of base formula: ${{\log }_{a}}b=\frac{{{\log }_{c}}b}{{{\log }_{c}}a}\,;$

8)$x>0$ and $y>0\,\Rightarrow {{\log }_{a}}xy={{\log }_{a}}x+{{\log }_{a}}y\,;$

9)$x>0$ and $y>0\,\Rightarrow {{\log }_{a}}\frac{x}{y}={{\log }_{a}}x-{{\log }_{a}}y\,;$

13)$x>0,y>0,a>0,b>0,a\ne 1,b\ne 1\,\Rightarrow \frac{{{\log }_{a}}x}{{{\log }_{a}}y}=\frac{{{\log }_{b}}x}{{{\log }_{b}}y};$

14)$x>0,a>0,a\ne 1,n\in N\,\Rightarrow {{\log }_{a}}x={{\log }_{{{a}^{n}}}}{{x}^{n}};$

$a>1$	$x\in \left( 0,1 \right)$	${{\log }_{a}}x<0$
$a>1$	$x>1$	${{\log }_{a}}x>0$
$0<a<1$	$x\in \left( 0,1 \right)$	${{\log }_{a}}x>0$
$0<a<1$	$x>1$	${{\log }_{a}}x<0$

15)

$a>1$	$0<x<y$	${{\log }_{a}}x<{{\log }_{a}}y$	The function is strictly increasing
$0<a<1$	$0<x<y$	${{\log }_{a}}x>{{\log }_{a}}y$	The function is strictly decreasing

16)$x\in R,a>0.a\ne 1\,\Rightarrow {{a}^{x}}={{e}^{x\ln a}}$

17) The logarithmic function is invertible and its inverse is the exponential function $f(x)={{\log }_{a}}x,$ ${{f}^{-1}}(x)={{a}^{x}},$ the two graphs are symmetric about the first bisector, the line $g(x)=x.$

Limits involving logarithms

$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{lo{{g}_{a}}x}{{{x}^{\alpha }}}=0$	$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{{{a}^{x}}}{{{x}^{\alpha }}}=\infty$
$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{lo{{g}_{a}}x}{{{a}^{x}}}=0$	$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{{{a}^{x}}}{lo{{g}_{a}}x}=\infty$
$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{\alpha }}}{{{a}^{x}}}=0$	$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{\alpha }}}{lo{{g}_{a}}x}=\infty$

These limits can also be read off the graphs above.

For a>1	For 0<a<1
$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,({{\log }_{a}}x)=\infty$	$\displaystyle \underset{x\to \infty }{\mathop{\lim }}\,({{\log }_{a}}x)=-\infty$
$\displaystyle \underset{\underset{x>0}{\mathop{x\to 0}}\,}{\mathop{\lim }}\,({{\log }_{a}}x)=-\infty$	$\displaystyle \underset{\underset{x>0}{\mathop{x\to 0}}\,}{\mathop{\lim }}\,({{\log }_{a}}x)=\infty$