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Written by Bryce Ringwood

Logarithms were once used to simplify multiplication and division. They do so by replacing them with the simpler processes of addition and subtraction and the use of lookup tables called log tables.

Although this is now archaic, logarithms are functions in their own right and as we shall see later in the sections on calculus, provide us with a tool for solving equations and modelling what goes on in the designs we are working on.

John Napier is regarded as the father of logarithms, with the publication of his tables in 1614. He seems to have based his calculations on the series for natural logarithms (taking a mammoth 20 years to complete the task.) Although his work caused many schoolchildren of my generation to despair, his achievement was a godsend to the scientific community and to professional engineers and surveyors.

The first pocket calculator was introduced in 1970 by Busicom, using the Intel 4004 chip. This device spelled the end of the widespread use of log tables. Even before that time, mechanical calculators like the FACIT were used by surveyors for accurate calculations.

I'm going to touch briefly on the use of logarithms as a device for performing arithmetic. Then we will look at logarithms in the 21st century, in particular we will look at decibels. Just for fun, we will look at the slide rule. I left this as the last section before we go on to calculus, because you need to know about "e"  (Euler's Number-described in the previous section) before you can really make sense of natural logarithms.

##### Explanation and use in Multiplication and Division Sums

If $\displaystyle{p = b^x}$ then for some number $\displaystyle{b}$:

$\displaystyle{log{_b}(p)=x}$ (We will use this result later)

In words, "log of p to the base b is equal to x". The base, "b" can be any number, but for our purposes, we will stick to 10 - giving us common  logarithms or "e"giving us  natural  logarithms.

On your HP calculator, TrayCalc and CalcPad, these are the LOG and LN keys. In BASIC,FORTRAN and 'C' the LOG() function returns the NATURAL log. There is NO LN() function. (Gotcha!).  FORTRAN has a LOG10() function for logs to the base 10.

Ouch.

Suppose now $\displaystyle{q = b^y}$,then: $\displaystyle{log{_b}(q)=y}$

We can also write $\displaystyle{p\times q = {b^x}{b^y} = {b^{x+y}}}$, so

$\displaystyle{{\color {red}{ log{_b}(pq)} }= {x + y} = \color {red}{{log{_b}(p) + log{_b}(q)}} }$

"The log of p multiplied by q = the log of p plus the log of q"

That is the magic of logs.

Example: to multiply 14.53 by 12.72, we first calculate their logs (use your calculator for this):

log 14.53 = 1.162266

= 2.266753   Now press the $10^x$ key to get the result or antilog =184.821..

You can check by simply multiplying that this is the right answer.

You might like to go through the same logic for the division of p by q. You can try the same sums using "e" as your base - you need to press the $e^x$ key to take the antilog. Of course, young children growing up in the '50s were taught logs at a very early age - much of the pain being the manipulation of log and antilog tables and sweating over "mean differences"to get an accurate result.

Using your calculator, what is $log{_{10}}(8.5)$ and $log{_e}(21.6)$ ?

##### Logarithms Today

Decibels

The human ear responds to a very wide range of sound pressures. In the middle of the night we can be woken up by a mosquito flying nearby. On another occasion we may have to endure very loud sounds from a rock concert (say). The comparative range of sound amplitudes could be over a million. This corresponds to the dynamic range of the human ear. Tests have shown that we perceive sound to be in equal increments of volume corresponding to a logarithmic scale. For example, if the rock concert has a sound pressure or amplitude of 1 million times the reference sound level and the mosquito produces a sound amplitude of 1  times the reference level, then the power ratio ratio could be expressed as ${1000000\times1000000}\over {1\times 1}$ - very inconvenient. (Squares are used for comparison of power, The amplitude ratio would be a mere million times).

Instead, we use a logarithmic decibel scale, and express the power ratio as $20{log({1000000\over1})} = 120dB$. Logs are to base 10. The amplitude ratio would be 60dB.

This explains why we use potentiometers with a logarithmic response (output voltage is proportional to the log of the shaft rotation) in amplfier volume controls.

Decibels are a very convenient way to express ratios. In electronics, we have the formula for power in a circuit as ${V^2}/R$, so we define decibels as power or voltage ratios as follows:

dB = 10 log{ {P_2}\over {P_1}} = 20 log {{V_2}\over{V_1} }

You might also hear radio hams saying "you're coming in 20dB over S9". The signal strength S9 is sort of agreed to be an antenna voltage of 100µV,so the incoming signal would be

\displaystyle{{V_2} = {{V_1}\times 10^{dB\over 20}}={100 \times 10 ^{20\over 20}}=1mV}

Log Function

The log() is a function, just as its antilog ${10^x} or {e^x}$ is a function of x. Here is a plot of the natural logarithm as a function of x. Plot of the Natural logarithm

x gets smaller and smaller - but the graph never touches the y-axis.

When we look at calculus, we will find uses for logs and antilogs $e^x$ in problem solving.

##### Changing the Base

We saw that some varieties of BASIC do not have a common log function - just natural logarithms. As we saw above, some calculations require we use logaithms to base 10 - so we have to convert them from base "e". Suppose we have a number "p" for which we want logaritms to base 10, then:

$\displaystyle{p = {10^x} = {e^y} }$ ...    (1)

$\displaystyle{ log{_{10}}(p)=x and log{_e}(p)=y }$ ...    (2)  from (1)

$\displaystyle{ {log{_e}({10^x}) = y} so: { xlog{_e}(10)=y} and {x = {y \over {log{_e}(10)}}} }$ ...    (3)

$\displaystyle{ {x = log{_{10}}(p)} ={{log{_e}(p)}\over{log{_e}(10)} } }$ ...    (4)

Now we can write a simple GW-BASIC progam for a decibel conversion program:

10 REM power / voltage db ratio program
20 PRINT "enter 1 for power, 2 for voltage ratio",
30 INPUT R
40 PRINT"enter power or voltage 2",
50 INPUT P2
60 PRINT "enter power or voltage 1",
70 INPUT P1
80 PRINT "Db =  ",10*R*LOG(P2/P1)/LOG(10)
90 STOP
100 END

run
enter 1 for power, 2 for voltage ratio    ? 2
enter power or voltage 2    ? 2
enter power or voltage 1    ? 1
Db =           6.020599
Break in 90
Ok
run
enter 1 for power, 2 for voltage ratio    ? 1
enter power or voltage 2    ? 7
enter power or voltage 1    ? 3
Db =           3.679767
Break in 90
Ok

##### The Slide Rule The iconic picture below from 1959 Astounding Magazine is a lot of fun - but as we know now, an anachronism. The illustration is for "The Pirates of Ersatz" and depicts a brigand boarding a spaceship with a slide-rule in his mouth. Nobody in 1959 thought there would be pocket calculators in slightly over a decade.

To be honest, I'm not sure that many calculators would be much use either, as the accuracy is normally about 9 digits - not enough for astro-navigation. The SHARP E500 works to 22 digits - if you ask it nicely, and Microsoft's calculator also works to an astounding, if not astronomical number of digits in scientific mode.

A slide rule is simply two scales that can slide past one another. If you take two plastic rulers, you can arrange them so that one scale slides past the other. Then if you position the zero of one scale at the "two" of the other, then read the value on the first scale at position 3, you will see the number 5 - because the numbers will have just added together.

A slide rule does exactly the same thing - but because the scales are logarithmic, the values multiply instead of add. Slide Rule Multiplication

In the image above, we are multiplying 1.3 by 2.5. The 1 is over the 1.3 on the lower scale, and the cursor is on 2.5 on the upper scale. On the lower scale, the result is under the cursor on 325. You know the answer is about "3" (1.5 X 2) so the answer is 3.25.

Slide rules were commonly made as 5-inch or 10-inch models. The 5-inch model was easy to put in your shirt-pocket and accurate enogh for most engineering problems. The illustration is from a 10-inch model. 20-inch slide rules were made (I have some) but they are just too cumbersome and not accurate enough for surveying calculations anyway. With a 10-inch slide rule, I guarantee our pirate would be "lost in space."

##### Summary
• Logarithms were originally devised to simplify multiplication and division in simple arithmetic.
• Logarithms are useful for expressing the ratios of large numbers. (Decibel scale for example.)
• Logarithms are often expressed to base 10 (Common Logs) or base "e" (natural logs)
• Base conversion is a simple matter of dividing the answer by the log of the base you want to the base you have got.
• The slide rule is a simple mechanical calculation device based on logarithms
• The logarithm is another mathematical function, just as sine is a funtion.

Finally, if you are automating calculations using a programable calculator or computer - be sure to consult the program reference manual concerning the log() function.

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