Category: Math

As Bob mentioned in a blog post, he and I are reading The Road To Reality by Roger Penrose. At our last guys’ night out, we talked about logarithms and I figured it would not hurt any of us to review a few basic math concepts. I won’t try and cover it all in this one post but will follow-up with additional posts in this new math category.

Powers and logarithms are two sides of the same coin. When you raise a number (called the base) to a power, you are expressing how many times you want the base multiplied by itself (ex. 4² means to multiply 4 by itself 2 times or 4×4=16). In the expression 4², 4 is the base and 2 is the power.

Likewise, when you take the logarithm (we’ll call that log for short) of a number, we are asking what power we need to raise a specified base to in order to get our original number. In the above example, log base 4 of 16 is 2 (written log₄16 = 2).

Since these two operations undo each other, we can combine them to do nothing but look like we are doing a bunch. For example, if we take the base 4 to the power 2 and then find the log base 4 of that result, we get log₄(4²) = 2. Since we put in 2, we get 2 out. As long as we do not change the base, the two functions cancel each other.

You might ask, “what good are logarithm?” Well, for years folks used logarithms to aid with doing tedious multiplication problems. Notice what happens when you take 2 numbers (say 16 and 64) and we take the log base 4 of each and add them together. In the first case we get log₄ 16 = 2 and in the second case we get log₄ 64 = 3 so our sum is 5. It also turns out that 4⁵ = 1024 which is exactly the same as 16 x 64 (our two starting numbers). We have just multiplied by adding! That is how a slide rule works (more on that in another post).