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Issue 9, 17 October 2022
By: Anthony O. Ives
Logarithims were briefly introduced in two previous blog articles, see Ref [1] and [2]. Originally logarithms were used for doing arithmetic with large numbers before the introduction of the electronic calculator. Such things as log tables and slide rules were used to do calculations in the way electronic calculators and computers are used today. Flight computers are devices that are used by pilots still today to do calculations. They are a circular slide rule and make use of logarithmic scales, see picture of modern flight computers below:
Logarithms are related to indices, see Ref [3] and can also be used to work out an unknown indice. Logarithims are generally known just as logs for shorthand.
\[2^2=4, log_{2} (4)=2\]
\[10^4=1000, log_{10} (1000)=4\]
\[10^1=10, log_{10} (10)=1\]
\[5^1=5, log_{5} (5)=1\]
\[e^1=e, log_{e} (e)=1\]
\[2^1=2, log_{2} (2)=1\]
Generally from the above if the small subscript number known as the base of the log is equal to number you take the log of the logarithim will equal 1.
\[ log_{10} (1000)=log_{10} (10^4)=4 \times log_{10} (10)=4\]
\[ log_{2} (4)=log_{2} (2^2)=2 \times log_{2} (2)=2\]
There are also ways to simplfy log expressions that are being added or subtracted however this can only be done when the log base is the same.
\[ log_{10} (100) + log_{10} (1000)\]
\[ =log_{10} (100 \times 1000) \]
\[ =log_{10}(100000) \]
\[ = 5\times log_10 (10) \]
\[ =5\]
\[ log_{10} (100) - log_{10} (1000) \]
\[ = log_{10}\frac{100}{1000} \]
\[ =log_{10}(0.1)\]
\[ =-1\]
However the most usefully function of logs is using them to find an unknown indice such as in the following example:
\[ 100^n = 10000 \]
\[ n = \frac{log_m10000}{log_m100} =2 \]
If you take log of both sides as above you can work out the unknown indice. It does not matter which base the log is just that obviously you use same log base on both sides. However the above is easier using log to base 10 without even needing a calculator.
This brings us to the two most used log bases which is e and 10, these two log bases are also common on most scientific calculators and computer programs. Log to base e is commonly refered to as natural logarithims. e is a random number 2.71828...... which is commonly associated with how naturally occuring systems decay or grow such bacteria growth, thermal cooling, etc. An example for aircraft of exponential decay is roll damping either of wing or rotor, when the pilot centres the control the roll rate decays at an exponential rate related to e. However all aircraft dynamics decay or increase at an exponential rate related to e depending on whether they are stable or not. Typically a decay or growth expression for naturally occurring system would look something like the expression below:
\[ y =Ae^{kt} \]
The term k is positive if its growth or negative if its decay. The A is representing amplitude and t represents time. The following graph show the expression plotted both for growth and decay scenarios.
Natural logarithm is typically represented as below:
\[ log_e x =ln x\]
Logarithim to the base 10 is typically represented as following:
\[ log_{10} x =log x\]
However sometimes loge can be represented by just simply log so you generally need to check what log represents is it base 10 or base e for the computer program or calculator you are using. The easiest way to test is to take log 10 does it give you 1 if not then it's most likely natural logarithm. The below shows typical logarithm functions on a scientific calculator:
Here is are summary of log arithmetic using general formulae expressions for logarithm relations.
\[x^1=x, log_{x} x=1\]
\[x^n=y, log_{x} y=n\]
\[x^n=y, n =\frac {log_m y} {log_m x} \]
\[log_m x^n= n log_m x\]
\[log_m x y= log_m x + log_m y\]
\[log_m \frac xy = log_m x - log_m y\]
Hopefully this article has been helpful in explaining logarithms and their functions as well their possible uses for solving problems. More information on logarithms can be found in Ref [4].
Please leave a comment on my facebook page or via email and let me know if you understand how logarithms are related to indices and their application for aircraft dynamics as well range and endurance calculations.
References:
[1] http://www.eiteog.com/No6EiteogBlogRange.html
[2] http://www.eiteog.com/No7EiteogBlogEndurance.html
[3] http://www.eiteog.com/No8EiteogBlogIndices.html
[4] Engineering Mathematics, K. A. Stroud, Fourth Edition, 1995, Macmillan
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