Geology Reference
In-Depth Information
(2)
Natural logarithm
= 'log to the base
e
' (where
e
=
2.71828, an important mathematical constant),
usually symbolized 'ln': ln(
X
) =
Z
where
X
=
e
Z
.
In words, the natural logarithm of a number
X
is
the power
Z
to which 2.71828 would have to be
raised to equal
X
. Since 100 can be expressed as
e
4.605
, ln(100) = 4.605.
δ
δ
y
t
So
=+
2
at at
δ
As
δt
→ 0,
dy
dt
=+=
202
at
at
(A7)
This indicates that the gradient at any specific point
t
1
on the curve can be calculated simply by evaluating
2at
1
.
The expression 2
at
is called the
derivative
of
y =
at
2
+ c
with respect to
t
. The algebraic or arithmetical
process by which we calculate the derivative is called
differentiation
, the basic operation in a branch of
mathematics called
differential calculus
.
By similar reasoning one can show that the deriva-
tive of
y
=
at
3
is 3
at
2
, and the derivative of
y
=
at
4
is 4
at
3
and so on. In general:
Logarithms also have the useful property of making
an exponential plot linear, as discussed below in rela-
tion to the Arrhenius plot.
To transform a logarithm (log
10
(
X
) or ln(
X
)) back into
the original number
X
, use:
()
()
log
X
ln
X
X
=
10
10
or
X e
=
.
if
y
=
bx
n
then
d
d
y
x
These calculations are easily done on a calculator or in
a spreadsheet.
When we want to plot a graph of a quantity that
varies over several orders of magnitude (e.g. from
0.02 to 5000), it may be useful to plot the logarithm
of the quantity while labelling the axis with the
original values, as in Figures 11.1, 11.2, 11.3 and 11.5.
A logarithmic axis like this can be recognized by
the appearance of
successive powers of 10
at regular
intervals.
nbx
n
−1
.
=
3
2
For example, if
yaxxxd
x
=+++
=
3 2
152 9 68 43
.
+
.
x
−
.
x
−
.
then
d
d
y
x
2
2
=
3
ax
+ += +−
2
bx c
456 8 68
.
x
.
x
.
To determine the numerical value of the gradient at a
specific value of
x
, one simply introduces that value of
x
into the
dy/dx
equation:
For example,
x
= 2.0:
d
d
y
x
=
Inverse square 'law'
182 4116 16 8
.
+ −=
.
.
177 2
.
.
The gravitational force between two bodies of mass
m
1
and
m
2
conforms to the equation:
Thus one can evaluate the slope of the curve at any
desired point.
For a more extensive introduction to differential cal-
culus, you should refer to the topic by Waltham (2000).
mm
r
(A8)
F
=
G
12
2
Logarithms ('logs') and their inverse
where
G
is the gravitational constant (=6.674 × 10
−11
m
3
kg
−1
s
−2
or N m
2
kg
−2
) and
r
is the distance between
the centres of gravity of the two bodies in metres. A
similar equation applies to the electrostatic force
between two electric charges
q
1
and
q
2
(e.g. between
electron and nucleus in an atom, as in Chapter 5):
Logarithms provide a handy means of plotting a func-
tion that varies over several orders of magnitude, com-
pressing it in a way that allows detail to be seen
throughout the range.
Logarithms used in this topic are of two kinds:
(1) '
Log to the base 10
': log
10
(
X
) =
Y
where
X
= 10
Y
. In
words, the log
10
of a number
X
is the power
Y
to
which 10 would have to be raised to equal
X
. Since
100 = 10
2
, the log
10
of 100 = 2.00. 4512 can be
expressed as 10
3.654
, so log
10
(4512) = 3.654.
Fk
qq
r
(A9)
=
12
2
e
where
k
e
is known as Coulomb's constant (=8.99 × 10
9
N m
2
C
−2
).
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