Logarithms

 

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 value into exponential form, say 1622: then

 

1622 = 103.21                           (1)

 

The log10 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 values that range over several orders of magnitude – say, from 1 to 100,000.  The log of these values only ranges between 0 and 5, since:

 

1 = 100 and therefore log(1) = 0

100,000 = 105 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 109 (based on the characteristic, we estimate x = 3.885 * 109.  Using a calculator, we get 3,887,340,000 – pretty darned close considering we only have 2 significant figures to deal with. 

 

Natural Logarithms

 

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