Graphics Reference
In-Depth Information
Thus a multiplication can be translated into an addition using logs:
36
×
24 = 864
log
10
36 + log
10
24 = log
10
864
1
.
55630250077 + 1
.
38021124171 = 2
.
93651374248
In general, the two bases used in calculators and computer software are
10 and 2
.
718281846
...
. The latter is e, a
transcendental
number. (A tran-
scendental number is not a root of any algebraic equation. Joseph Liouville
proved the existence of such numbers in 1844.
π
, the ratio of the circumfer-
ence of a circle to its diameter, is another example.) To distinguish one type
of logarithm from the other, logarithms to the base 10 are written as log, and
logarithms to the base e are written as ln.
From the above notation, it is evident that
log(
ab
) = log
a
+log
b
(3.16)
log
a
b
=log
a
−
log
b
(3.17)
log(
a
n
)=
n
log
a
(3.18)
log(
√
a
)=
1
n
log
a
(3.19)
The following formula is useful to convert from the base 10 to the base e:
log
a
=ln
a
×
log
e
=0
.
4343ln
a
(3.20)
3.6 Further Notation
Mathematicians use all sorts of symbols to substitute for natural language
expressions. Here are some examples:
<
less than
>
greater than
≤
less than or equal to
≥
greater than or equal to
= approximately equal to
≡
equivalent to
=notequalto
For example, 0
1 can be interpreted as: 0 is less than or equal to
t
,
which is less than or equal to 1. Basically, this means
t
varies between 0 and 1.
≤
t
≤
3.7 Summary
The above description of algebra should be su
cient for the reader to under-
stand the remaining chapters. However, one should remember that this is only
the beginning of a very complex subject.
