A Simple, Practical Explanation of Logarithms and Converting Number Systems

If you’re anything like me, you may have had to do logarithms in high school or college and were told how to do the formulas. Move the x over there, the base number here, and so forth. However, you were never told what they were or when you would use them.

In fact, I never understood logarithms until I began studying live sound engineering so that I could better serve the church that I attended as an apprentice, if you would. So, in my explanation of logarithms, I will illustrate the realizations that I came across while studying sound engineering. If I’m wrong, please let me know.

How the cochlea works

The cochlea is what reports sound frequencies to your brain and looks like a wrapped up snail. If you were to unravel it, it would look like this (pardon the ugly):

See how the number of Hz increases per distance down the cochlea? This is a logarithmic increase. If the increase were linear (the same distance for every Hz), the drawing would go WAY off the edge of the page. If you remove the cochlea part of the ugly (the drawing), this is the same scale that sound engineers look at when working with frequencies

Where do you think 500 Hz would go? It’s not in the middle of 100 and 1,000. It’s more like here:

This means that the same increase in perceived pitch (how far you move down the cochlea drawing) requires a greater number of Hz each time.

Further explanation of this concept

Your body perceives (pretty much) everything logarithmically. Take your eyes, for example. To perceive the same increase in brightness from say, 1x to 2x, you need to double 1x, or double the number of photons. However, to perceive the same increase in brightness from 2x to 3x, you need to double 2x, not 1x, so the total number of photons is actually 4 times what they were at 1x. This is a logarithmic increase.

To perceive the same increments in hue, or light frequency, you need to double the frequency of the light every time.

What about sound? Have you heard of deciBels (dB)? You’ve probably heard that a plane engine is 120dB, but what does that mean? Turns out, you perceive volume, or the amount of pressure exerted on your eardrums, logarithmically, meaning that for every perceived incremental increase in volume requires a doubling of the intensity, or the force applied to your eardrums.

A deciBel is 1/10th of a bel (deci = 1/10th of something). Every 10 dB, or one Bel, is a doubling of the amount of pressure on your ear. This means that 60dB is twice as loud as 50db and 70db is twice as loud as 60dB. Because you perceive volume (pressure) logarithmically, you only hear these every increasing amplitudes at the same increments of loudness.

A more practical illustration

On a piano, the international standard of A is 440Hz. An octave, or 8 white keys on either side until you get to a key of the same letter, is a doubling or halving of the frequency. An ugly illustration that probably uses the wrong shape for the keys:

However, if you have the chance to hit the keys, you will perceive the same increase in pitch as you go along. This is due to your cochlea being logarithmic, just like the piano’s keys.

That’s great, but you haven’t told me what a logarithm is

A logarithm is, in essence, something that requires the same kind of increments that we’ve talked about.

What the heck is the base, though? Have you heard something like, “log base 10 of 8?” What is that base? My teacher only told us to use 10. Why? She really didn’t know.

The log’s base is the base of the numbering system that you use. We use base 10, or 0-9, which has 10 units. If we use binary, 0′s and 1′s only, it’s base 2. Hexadecimal, 0-9 and A-F, is base 15. Before the Arabic numbering system was adopted by Europeans, they used, I believe, a base 12 system. The Babylonians, I’ve read, used base 60 (why!?).

Do you remember in elementary school how you may have had to do exercises like:
245 =
2 x 100 +
4 x 10 +
5 x 1

You use this to convert between numbering systems. Let’s say that an alien race, who has a base 5 numbering system, says that have 245 ships with which to invade Earth. How many is that? Let’s map out the places. (note that it’s not possible to put 6 into five. You have to move into the next “slot” or place”)

25  5  1
2    4  5
2 x 25 = 50
4 x 5 = 20
5 x 1 = 5 
75 ships

So, if this alien race were to do logarithms, they would use log base 5.

So, what do logarithms do?
Logarithms go further than the general idea that I’ve explained here, which is that an equal increase requires greater input. This is where my practical knowledge ends (or, at least what I think I know). Continuing on is a good place for someone to take something that I’ve noticed and make an easy way for people to correlate this concept into what we’re doing when we solve logarithms for numbers.

For the mathy part, here’s a pretty good explanation to basic logs that work out perfectly and easily (like they always do in text examples but not the homework assignments)

A summary of the page is “How many times do you need to multiply a number by itself to get another number.”

An interesting correlation
I’ll be honest, most of the math is a bit beyond me. This is the part where it would be good for someone to take this concept and make it easily transferable to numbers.

However, I did notice a correlation between the number conversation’s and logarithms.

Take, for example, log2(8)=3, which means that you have to multiply 2 by itself 3 times to get 8.

If you use the long addition way of converting numbers like we did above, you get

3 2 1 0 [position index, starting at 0, like we do in programming]
8 4 2 1 [the number base, in our case, binary, which is from log base 2]
1 0 0 0

The number of times that you need to multiply 2 by itself to get 8 is where the 1 (or result?) of the binary conversion is (at the base 0 index).

Another example with perfect numbers (from the link above)

4        3      2    1   0  [index of the place, starting at 0]
625 125   25   5   1  [the number represented by each position]
1        0     0     0   0  [what 625 looks like in base 5]

4 matches up with 1 in the highest place holder.

Let’s try again with another number that isn’t perfect and results in a non-whole number and where my brain melts, like our 245 ships using the base 5 numbering system.

log5(245)=3.418 (from wolfram alpha)

3       2     1    0 [the position index, starting at 0]
125  25   5     1 [the number base]
1       4     4     0

It’s getting to be a bit late for me and my non mathy brain is having trouble mapping the indices and positions to the decimal version of how many times you need to multiply 5 by itself to get 245.

What I mean is that, on the bottom line, the 1 corresponds to 3. The rest appears to be fractions that should (if my idea is correct) equal the .418 of 3.418.

Can someone smarter than me finish this or let me know if I’m way out in left field?

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