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Back before electronic calculators, there were logs (logarithms) and Slide Rules. Logarithms were invented by John Napier. They were a way to multiply by adding. "Why not just multiply?" you may ask. Well sometimes logs were easier, especially with powers and roots. Before we find out what logs are, let's see that we can multiply by adding:
2 x 3 = x?
log 2 = .30103 (which I looked up in a table)
log 3 = .47712
log x = log2 + log3 = .77815
log 6 = .77815 (also in the table)
Of course we knew the answer was 6 already. But, you see how it works. Let's try something tougher. What is 7.3^6.4? You may have never seen fractional powers. x^0.5 is the square root of x. 7.3^6.4 is 7.3^6 x 7.3^2/5 or 7.3^6 x the fifth root of 7.3 squared. Well the log of 7.3 is .86332 and log 7.3^6.4 is 6.4 times that (as you will see below) or 5.52525. Looking that up in a table, I find that 7.3^6.4 is about 335100.
Logs are exponents (powers). Here are some rules about exponents (which should show you how to use logs):
Above, we have been using powers of 10. Let's continue to do that. 1 is 10^0 and 10 is 10^1. All of the numbers between 1 and 10 are fractional powers of 10. 2 has a log of .30103. That means that 2=10^.30103. That's all there is to it. Logs are powers. Above we multiplied 2x3=6. This is the same as 10^.30103 x 10^.47712=10^.30103+.47712=10^.77815=6.
I also got 5.52525 as a log. But my log tables only show logs from 0 to 1. 10^5.52525=10^5 x 10^.52525. I looked up .52525 in my log tables, and I got 3.35100. Multiplying that by 10^5 gives 335100. Your scientific calculator can show more accuracy than that. But that is pretty good anyway.
There are other bases, besides 10, for logs. In computers, we sometimes use a base of 2. The other popular base is e (2.71828...) (see e). We abbreviate a base 10 logarithm as log; we abbreviate a base e logarithm as ln (ln means natural logarithm). 0.069315 is the ln of 2. The arithmetic is more complicated, for numbers beyond e. So we don't use base e logarithms for arithmetic. But many results in calculus are greatly simplified by using base e.
Anyway, just remember that logs (logarithms) are exponents.