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How are logarithms used?
Many natural phenomena increase in a logarithmic way. For instance, earthquakes are described using the Richter Scale, a scale that is logarithmic. A magnitude 5 earthquake is 10 times larger than a magnitude 4 because the number is an exponent in a logarithm.
What are logarithms?
A logarithm is a way of writing a number that takes advantage of the properties of exponents. In this subject, a number like 1000 is considered as 103 which means 10 x 10 x 10. The lower number, (10) is called the base, and the upper number the exponent.
To 'take a logarithm' of a number means to write the exponent, usually with a base of 10 (although other bases can be used). So, the logarithm of 10,000 would be 4 and comes from the relationship 10,000 = 104. This is also written as log(10,000) = 4.
The real benefit of logarithms comes from the fact that when numbers with exponents are multiplied together, the exponents add up. For instance, 102 times 103 = 105, because 2 + 3 = 5. In logarithms, the same process is used - large numbers can be multiplied simply by adding up their logarithms.
Numbers that aren't exact multiples of 10 also have logarithms, so that multiplying using logarithms is possible with any number.
A basic problem with logarithms.
To multiply any two numbers with logarithms, you first take the log of each number, then add them and then take the anti-log (an anti-log is the opposite of a log, it takes a logarithm and converts it back to a number).
Multiplying 123456 and 456789 together would look like this:
Log (123456) = 5.091512 (this is approximate, so our final answer will also be approximate). Log (456789) = 5.659715 and adding these two logarithms together gives 10.751227.
Our final answer is 1010.751227 which is about 56393233880.
If you use a calculator and multiply the two numbers directly, you get:
123456 x 456789 = 56393342784, the exact answer. Although in this example, I had enough digits on my calculator to get a precise answer, for really big numbers, logarithms are sometimes the best method.
An interesting fact about logarithms.
As computers become faster and faster, other methods for handling calculations with large numbers have replaced this use of logarithms. But the relationships logarithms describe seem to be embedded in nature.
Human eyes sense how bright an object is on a logarithmic scale, so that to notice one thing is brighter than another, a certain logarithmic threshold has to be passed. As objects get brighter overall, you need even more of a change to notice a difference.
This is why star magnitudes are based on logarithms. Ancient astronomers ranked our sun as one (the first magnitude) and the dimmest star they could see as six. Each celestial object between these extremes was assigned a magnitude by how bright they seemed. The scale is logarithmic because a doubling of brightness only moves a single step on the scale.
Who invented logarithms?
Logarithms and the methods for their use were published in 1614 by Baron John Napier, of Scotland. They found a great deal of use in astronomical calculations and since then, in many other areas of science. The pH scale in chemistry (a measurement of acidity) is based on logarithms and there are many others.