Logarithms

Definition: Let and 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 , ,

clearly

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

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)

2)

3)

4)

5)

6)

7. Change of base formula:

8)$x>0$ and

9)$x>0$ and

13)

14)

$a>1$
$a>1$	$x>1$
$0<a<1$
$0<a<1$	$x>1$

15)

$a>1$	$0<x<y$		The function is strictly increasing
$0<a<1$	$0<x<y$		The function is strictly decreasing

16)

17) The logarithmic function is invertible and its inverse is the exponential function the two graphs are symmetric about the first bisector, the line

Limits involving logarithms

These limits can also be read off the graphs above.

For a>1	For 0<a<1