Logarithms are one of the most useful tools in pure and applied mathematics. They are used throughout all of science and engineering, and understanding them is critical for any scientist or engineer. They are also a cause fear for some students. This article tries to explain them simply.
Logarithms - What Are They? The Shorthand of Mathematics
Logarithms are not a stand-alone, separate idea. They are part of a series of "convenient ways of writing". The first "summary" is actually multiplication:
1 + 1 = 2
But what about
1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 9 ?
It is true that the sum is 9, but it is hugely inconvenient to write. What if the total was 1,000 or 1 million? A way needed to be found to write down the question in summary form. So the multiply ("×") sign was used:
2 × 1 = 2, and
9 × 1 =9
By the same token, it is fairly convenient to write
2 × 2 = 4, but what about
2 × 2× 2× 2× 2× 2× 2× 2× 2 = 512 ?
Methods to show this were invented too. The caret ("^") symbol is used where superscripts are unavailable:
2^9 = 512
In words, this is "two to the power of nine equals five hundred and twelve". The logarithm in this instance is defined as the power to which 2 must be raised to get 512, or "how many times must 2 be multiplied by itself to get 512.
Logarithm Properties And Rules
Using the previous example
2 × 2× 2× 2× 2× 2× 2× 2× 2 = 512, we can derive one of the basic rules of logarithms:
What is 2^3 × 2^6 ?
(2 × 2× 2)× (2× 2× 2× 2× 2× 2) = 512, or 2^9, so
2^3 × 2^6 = 2^9, or
2^(3 + 6) = 2^9
The same can be shown for 2^1 × 2^8, or 2^2 × 2^7 etc. In other words, it doesn't matter which groups are put into brackets - the result is the same.
This very important result, simply explained, leads to the best-known rule with logarithms:
log(x) + log(y) = log(x × y)
It is this result that allows large numbers to be multiplied or divided using logarithms.
When teaching logarithms to students who are seeing them for the first time, it is useful to get them to write out the long-hand method first (e.g. 2 × 2× 2× 2× 2× 2× 2× 2× 2) . Doing this with at least five examples, and then allowing them to write 2^9 etc, really shows the benefits of writing the short-hand version. This also de-mystifies the exponent symbol.
Logarithm Summary
Logarithms hold fear for many students - but they should not. They are nothing more than a short-hand way to write down something that would otherwise take too much time and space. This article explains why the exponent (power) rules and the rules of logarithms are true, and why they should be easy to understand.
It is also noteworthy that the first differential of log(x) is 1/x. (This is true when the logarithm base is "e", the sum of the series
1 + 1/1 + 1/2! + 1/3! + ...
where 3! (3 factorial) is 3 x 2 x 1).
Logarithm References
" Math Made Nice & Easy #2: Percentages, Exponents, Radicals, Logarithms and Algebra Basics"
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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