We use log functions a lot in biochemistry, so it’s useful
to remember a few things about logarithms.
**Warning to mathematicians**: this is written by a biochemist for
biochemistry students. If you see
anything that’s out-and-out wrong, please let me know. Otherwise please remember that this is a
practical discussion for practical purposes.

**Common (Base 10) Logarithms:**

If we put some random

1622 = 10^{3.21} (1)

The log_{10} function (usually just *log*) is
such that:

log(1622) = 3.21 (2)

There are several conveniences to the use of logs. In the first place, we are frequently
interested in

1 = 10^{0} and therefore log(1)
= 0

100,000 = 10^{5} and therefore log(100,000)
= 5

The pH scale of 0-14 (pH is –log([H^{+}]))
thus represents a range of [H^{+}] concentrations from 1 M to
0.00000000000001 M – much more convenient.

In order to manipulate logs arithmetically, it is important
to remember that they are exponents. The
integral part of a logarithm is the “**characteristic**;” the decimal
portion is the “**mantissa**.” From
equation (2) above, the logarithm of 1622 is 3.21. Therefore, 3 is the characteristic and 0.21
is the mantissa.

The nature of exponents makes log manipulations simple:

_{} (3)

_{} (4)

_{} (5)

_{} (6)

In the “olden days” before calculators, people would have tables of logarithms from which you could get the mantissa. An example is appended below. The characteristic could be determined by eyeball (just count places to the left of the decimal point); the mantissa could be obtained from the table. For example, 1622 is 1000 * 1.622 – the characteristic 3 corresponds to the 1000; the mantissa 0.21 is the log of 1.622. Changing the order of magnitude of a number changes the characteristic but not the mantissa:

log(1622) = log(1000) + log(1.622) = 3 + 0.21 = 3.21

log(16,220) = log(10,000) + log(1.622) = 4 + 0.21 = 4.21

log(162.,200,000,000,000 = 14 + 0.21 = 14.21

log(0.01622) = log(0.01) + log(1.622) = -2 + 0.21 = -1.79

As an example, consider the following:

x = 402.,000,000,000,000 * 0.00000967

well, the characteristic of 402.,000,000,000,000 is 14 and, from the table below, the mantissa for 4.02 is 0.6042261, so log(402.,000,000,000,000) = 14.6042261; log(0.00000967) = log(0.000001) + log(9.67) = -6 + 0.9854265= -5.0145735

log(x) = log(402.,000,000,000,000) + log(0.00000967) = 14.6042261 – 5.0145735 = 9.5896526

Reading backwards off the table, looking up the mantissa
0.5896526, we get a number between 3.88 and 3.89. Multiplying by 10^{9} (based on the
characteristic, we estimate x = 3.885 * 10^{9}. Using a calculator, we get 3,887,340,000 –
pretty darned close considering we only have 2 significant figures to deal
with.

Natural logs are handled exactly like common logs. The difference is in the base. Common logs represent powers of 10; natural
logs represent powers of **e**. **e** is an irrational number, approximately equal to
2.718282. **e**
isn’t just arbitrary – it is the number with the rather fascinating
mathematical property that:

_{} (7)

The “e to the” function is generally referred to as the
exponential function, usually abbreviated **exp**. So the statement

_{} (8)

is the same as

_{} (9)

The natural logarithm, **ln**,
is the number such that if _{}, then _{}. There is a simple
rule of thumb for converting between ln and log: Since, by the nature of the definition of the
logarithm, for any number x (x ≠
0),

_{} (10)

taking the natural log of both sides:

_{} (11)

and therefore from (5) above:

_{} (12)

and since _{}

_{} (13)

Appendix
– Table of Common Logs