Exponents, Radical and Logarithms
I’ve just finished a series of short posts that goes into the etymology behind these terms, and now I’d like to take a quick look at the history and use cases for these operations.
The term “exponent” was first seen in the 16th century, in a work titled Arithmetica Integra (1544) written by Michael Stifel.
But the concept of squaring numbers and raising them to higher powers goes back much farther than that:
The earliest known symbols to indicate the power of a number appear in ancient Babylon and Egypt. It seems the Babylonians had ideographs for both squaring and cubing a value. In an Egyptian papyrus now in Moscow, a symbol, which looks like a drawing of a pair of walking legs, is used to indicate multiplying a value by itself. (Source)
Exponents are simply shorthand for the operation of multiplying a value by itself. Instead of writing 6x6x6x6x6x6x6x6x6x6, you can just write 6^10.
The shorthand for exponentiation has changed over the centuries, until arriving at our modern-day usage.
Here are some examples of exponentiation in real life (Source):
1. Measurement. More specifically, area and volume. A square foot is the area of a square whose sides are equal to 1 foot. Area is lengthxwidth, and if all the sides are the same we get 2x2 which is 2^2. Similarly, a cubic meter measures the volume of a cube with sides that are equal to 1 meter. Since a cube has three sides, we get 3x3x3 which is 3^3. I also like how the 2-dimensional measurement is squared, and the 3-dimensional measurement is cubed.
2. Bits and bytes. Computers are based on binary architecture, and multiples are expressed in powers of 2. Thus we have 32-bit and 64-bit computers, download speeds that jump from 128kbps to 256kbps, and so on and so forth.
3. Growth. Exponential growth is a crucial component of demographics (i.e. the rate at which a population grows), computing (i.e. the rate at which the speed and power of computers increases), finance (i.e. the rate at which money invested earns interest on the interest), biology (i.e. the rate at which diseases spread) and other natural sciences (i.e. the rate at which radioactivity decays).
The radical is another shorthand notation. It is the symbol by which we acknowledge that the radicand (the number underneath the repeat bar) represents a value that has been multiplied by itself a given number of times (root).
Finding the root of a power can be a complicated process, as the answer frequently requires decimal notation. But why would we need to find the root number in the first place? When would we ever use this?
1. Construction. Builders may need to perform a “diagonal check” on the building to ensure that the final structure will hold the desired shape. They wouldn’t be able to perform this check without knowledge of square roots. (Source)
2. Finding outliers. The normal distribution is used all over the place to analyze populations and the distribution of variables throughout that population. If you’re interested in finding “abnormal” or anomalous values, then you’ll need to know square roots, as the equations for finding these values use this operation. (Source)
3. Measurement. Since finding the root is the inverse of exponentiation, we can reverse our area measurement to find the dimensions of, say, an apartment. If you know the total area is 400 sq ft, then you can use the square root to figure out that the apartment is 20 ft x 20 ft (Source). You also need square rooting to find distance, especially when using the formula derived from the Pythagorean theorem.
Remember, the logarithm is the number to which a base is raised to get the power. We use the base and the power to find the exponent.
There is a basic logarithm called the “common logarithm” that is understood to operate on base 10. From what I understand, this operation was used to reduce the amount of time needed to perform calculations in science and engineering, for instance.
Here’s an example (from Source):
Historically, logarithms are interesting because they allow you to use addition to do multiplications. For example, if you want to multiply 365.49 by 1474.3, you can do this: x = 365.49 * 1474.3 log (x) = log (365.49 * 1474.3) 10 10 log (x) = log (365.49) + log (1474.3) 10 10 10 log (x) = log (10^2 * 3.6549) + log (10^3 * 1.4743) 10 10 10 log (x) = 2 + log (3.6549) + 3 + log (1.4743) 10 10 10 log (x) = log (10^5) + log (3.6549) + log (1.4743) 10 10 10 10 10 Now, that's cool, because you can look up the logarithms of the numbers 3.6549 and 1.4743 from a table; add them; and then look up the inverse logarithm of the result. Then multiply by 10^5 (by shifting the decimal place) to get the final product. This isn't such a big deal now, when cereal boxes give away calculators that can do it; but it's what made slide rules possible, and before that, it made a lot of calculations easy that would have been much more difficult.
Math Dr. Ian goes on to explain why logarithms are useful for finding unknown quantities, by showing how they are analogous to inverse-multiplication (i.e. division):
Think about what makes division useful. There's nothing you can do with division that you can't do more clumsily using multiplication in conjunction with guess-and-check. When you just want to multiply things, 16 * 48 = ? that's easy. When you only have one product and a result, 17 * ? = 592 division lets you invert the multiplication: 17 * ? = 592 -> 592 / 17 = ? Logarithms do the same sort of thing for exponents. Sometimes you just want to raise something to an exponent, and that's easy: 12^4.9 = ? But sometimes you have the base and the result, and you need to find the exponent: 13^? = 6.3 You _could_ figure this out with guess-and-check; but logarithms let you get the result directly: 13^? = 6.3 -> log (13^?) = log (6.3) 13 13 ? = log (6.3) 13
The common logarithm in particular is so important because it operates on base 10, and so many of our concepts operate on powers of 10. For example:
Better Explained tells us that logarithms help to put numbers on “human-friendly” scales like the above.
The trick to overcoming "huge number blindness" is to write numbers in terms of "inputs" (i.e. their power base 10). This smaller scale (0 to 100) is much easier to grasp:
power of 0 = 100 = 1 (single item)
power of 3 = 103 = thousand
power of 6 = 106 = million
power of 9 = 109 = billion
power of 12 = 1012 = trillion
power of 23 = 1023 = number of molecules in a dozen grams of carbon
power of 80 = 1080 = number of molecules in the universe
A 0 to 80 scale took us from a single item to the number of things in the universe. Not too shabby. (Source)
So how do we actually use this concept in real life?
1. To denote salary. We do this all the time! Whenever someone says they make a “6-figure salary,” they’ve used the common logarithm. According to Better Explained, “n-figure salary” is how we explain numbers in terms of their digits, that is, “how many powers of 10 they have (are they in the tens, hundreds, thousands, ten-thousands, etc.). Adding a digit means ‘multiplying by 10.′” This allows us to get “a rough sense of scale without jumping into details.” (Source)
2. Listening to music. “In the book The Joy of X – Steven Strogatz writes: Notice something magical here: as the numbers inside the logarithms grew multiplicatively, increasing tenfold each time from 100 to 1,000 to 10,000, their logarithms grew additively, increasing from 2 to 3 to 4. Our brains perform a similar trick when we listen to music. The frequencies of the notes in a scale – do, re, mi, fa, sol, la, ti, do – sound to use like they’re rising in equal steps. But objectively their vibrational frequencies are rising by equal multiplies. We perceive pitch logarithmically.” (Source)
3. The natural log and e. This is something that I’ll have to leave for its own post.
