If in the equation gx = a the values of x and g are known, you are confronted with a power, if the values of x and a are given, with a root. The third case when g and a are given and x is sought must still be discussed. No x is no more called an exponent, but the logarithm of the number a with the base g. Definition:
The
logarithm of the number a
with the base g is the
power
to which one must raise the base g
in order to obtain a.
Examples:
| if 2x = 8 | then x = 3 | if 3x = 9 | then x = 2 | etc. |
Since taking a logarithm is the inverse of taking a power and we have introduced for every inverse mode of calculation - recall subtraction, division, root taking!- a new symbol, we will do so again here.
For gx
= a
write
and say: The logarithm to the base g equals x. Thus, for
| 4x=16, x=2 | we will write |  |
||
| 2x = 8, x = 3 | " |  |
 |
 |
 |
etc. |
Terminology: In or gx = a, x
is called the logarithm, a is the argument and
g is the base of the logarithm.
By the definition above, 
is the
number to which g must be raised to obtain a.
Definition:
.
You add the base g to the symbol log whenever different bases may arise. As a rule, you let g be constant and write simply log a.
Note the following remarkable values:
| log 1=0 | since | g0=1 | log g=1 | since | g1=g | log gx=x | since | gx=gx |