= log (10-1)
= -1
Two types of logarithms are used in this
text: (a) common logarithms (abbreviated log) whose base is 10
and (b) natural logarithms (abbreviated ln) whose base is e (=2.71828).
log x = n, where
x = 10n
ln x = m, where x
= em
Most equations in chemistry and physics
were developed in natural, or base e, logarithms, and we
follow this practice in this text. The relation between log and
ln is
ln x = 2.303 log x
Despite the different bases of the two logarithms, they are used in the same manner. What follows is largely a description of the use of common logarithms.
A common logarithm is the power to which
you must raise 10 to obtain the number. For example, the log of
100 is 2, since you must raise 10 to the second power to obtain
100. Other examples are
log 1000 = log (103) = 3
log 10 = log (103) = 1
log 1 = log (100) = 0
= log (10-1)
= -1
=
log (10-4)
= -4
To obtain the common logarithm of a number
other than a simple power of 10, you must resort to a log table
or an electronic calculator. For example,
log 2.10 = 0.3222, which means
that 100.3222 = 2.10
log 5.16 = 0.7126, which means
that 100.7126 = 5.16
log 3.125 = 0.49485, which means
that 100.49485 = 3.125
To check this on your calculator, enter the number, and then press the "log" key. When using a log table, the logs of the first two numbers can be read directly from the table. The log of the third number (3.125), however, must be interpolated. That is, 3.125 is midway between 3.12 and 3.13, so the log is midway between 0.4942 and 0.4955.
To obtain the natural logarithm ln of
the numbers shown here, use a calculator having this function.
Enter each number and press "ln."
ln 2.10 = 0.7419, which means that e0.7419 = 2.10
ln 5.15 = 1.6409, which means that
e1.6409 = 5.16
To find the common logarithm of a number
greater than 10 or less than 1 with a log table, first express
the number in scientific notation. Then find the log of each part
of the number and add the logs. For example,
log 241 = log (2.41 x 102) = log 2.41 + log 102
= 0.382 + 2 = 2.382
log 0.00573 = log (5.73 x 10-3) = log 5.73 + log 10-3
= 0.758 + (-3) = -2.242
Significant Figures and Logarithms
Notice that the mantissa - defined as the decimal portion of the logarithm, in this case it is equal to 0.242 - has as many significant
figures as the number whose log was found. (So that you could
more clearly see the result obtained with a calculator or a table,
this rule was not strictly followed until the last two examples.)
Obtaining Antilogarithms
If you are given the logarithm of a number,
and find the number from it, you have obtained the "antilogarithm,"
or "antilog," of the number. Two common procedures used
by electronic calculators to do this are:
Procedure A Procedure B
1. Enter the log or ln. 1. Enter the log or ln.
2. Press 2ndF. 2. Press INV.
3. Press 10x or ex.
3. Press log or ln x.
Test one or the other of these procedures with the following examples:
1. Find the number whose log is 5.234.
Recall that log x=n, where
x=10n. In this case n=5.234. Enter
that number in your calculator, and find the value of 10n,
the antilog. In this case,
105.234=100.234
x 105=1.71 x 105
Notice that the characteristic (5) sets
the decimal point; it is the power of 10 in the exponential form.
The mantissa (0.234) gives the value of the number x.
Thus, if you use a log table to find x, you need only look
up 0.234 in the table and see that it corresponds to 1.71.
2.
Find the number whose log is -3.456.
10-3.456=100.544
x 10-4=3.50 x 10-4
Notice here that -3.456 must be expressed
as the sum of -4 and +0.544.
Mathematical Operations Using Logarithms
Because logarithms are exponents, operations involving them follow the same rules as the use of exponents. Thus, multiplying two numbers can be done by adding logarithms.
log xy=log x + log
y
For example, we multiply 563 by 125 by
adding their logarithms and finding the antilogarithm of the result.
log 563 = 2.751
log 125 = 2.097
log xy = 4.848
xy = 104.848 = 104 x 100.848 = 7.04 x 104
One number (x) can be divided by
another (y) by subtraction of their logarithms.
For example, to divide 125 by 742,
log 125 = 2.097
-log 742 = 2.870
Similarly, powers and roots of numbers
can be found using lograithms.
log xy = y(log
x)
As an example, find the fourth power of
5.23. We first find the log of 5.23 and then multiply it by 4.
The result, 2.874, is the log of the answer. Therefore, we find
the antilog of 2.874.
(5.23)4 = ?
log (5.23)4 = 4 log 5.23 = 4 (0.719) = 2.874
(5.23)4 = 102.874
= 748
As another example, find the fifth root of 1.89 x 10-9.
The answer is the antilog of -1.745.
(1.89 x
10-9)1/5=10-1.745=1.80
x 10-2