As we became proficient in mathematics we noticed that some of the operators had inverse effects on numbers. For instance, subtraction was the inverse of addition thus you could take 10+4 = 14 and reverse the effect with subtraction; 14 -4 = 10. Next, they looked for an inverse to multiplication and came up with division. When it came to exponentiation however they had to search for a while for its inverse.
Finally, in 1614 John Napier published a book entitled Mirifici Logarithmorum Canonis Descriptio (Description of the Wonderful Rule of Logarithms). The logarithm is the inverse of the exponent, where 10 to the 3rd power is 1000, the logarithm base 10 of 1000 is 3. Logarithms had several interesting properties.
If you added two logarithms together you obtained the logarithm of the number that was the product of the two original numbers. For example, the logarithm base 10 of 100 is 2 and the logarithm base 10 of 1000 is 3. If we add the two logarithms, 2 + 3 = 5, we find that 10 to the 5th power is 100,000 as is 100 * 1000. Similarly, when you subtract two logarithms you get the logarithm of the number that you get when you divide the two original numbers; 5-2 = 3 thus 100,000 / 100 = 1000. This fact allowed for the invention of the slide rule, which was what all the geeky engineering types used before they had computers.
I’m going to generate some figures to illustrate this next feature of logarithms but in the mean time, bear with me as I describe it. A plot of an exponential function on a standard Cartesian grid starts with a very shallow slope which increases rapidly until it has a very steep slope that approaches infinity. When you plot the same function on a grid where the x axis is graduated in logarithmic intervals, the exponential curve becomes a straight line. This makes it easy to identify exponential functions when you don’t know their formula. You plot the raw data on a logarithmic grid and if you see a straight line, the function is exponential.
Like I said, I’ll add some figures to illustrate what I’m talking about tomorrow and we will continue on our adventure.
Sweet dreams, don’t forget to tell the people you love that you love them, and most important of all, be kind.