So, I have this math exam Tuesday afternoon, and logarithms are making me think of the Cookie Monster song. And maybe cry a little.
I’ve searched the forums, read the posts and links, and it still all looks like gibberish. I keep reading and reading, thinking it’ll make sense if I just read it enough times, but I’m just reading the words without knowing what it means. I mean… ok… 10 log 3? Cool, I can put those numbers in my calculator, but why should I, and why does it say 4.77 when I do? Could anyone dumb it down enough that my head /doesn’t/ spin? Or at least try, please?
C is for cookie, that’s good enough for me. Cookie starts with C!
Is it just the definition of logs that is giving you trouble? Think of logarithms is being the “reverse” of doing exponents.
In other words, log(3) is the number that, when you raise 10 to that power, you get 3. Since log(3) is approximately .477, this is saying that 10[sup].477[/sup] = 3 (approximately. If we didn’t have to round the decimal numbers off, everything here would be exact).
It may be easier to see this way:
Take log(10[sup]7[/sup]). This is asking, “What power of 10 will give you 10[sup]7[/sup]?” I think it should be clear that the power we’re looking for is 7, itself, so log(10[sup]7[/sup])=7.
From that example, it’s hopefully clear that log(10[sup]x[/sup])=x for any real number x.
That’s just a basic introduction. If any of this is not clear, or if you want to go further with it, just ask.
While logarithms are still important, to understand them it’s useful to know why they were introduced.
Back it the years B.C. (before calculators), people did calculations using pen/pencil and paper, and it was a lot easier to do additions and subtractions than to do multiplies and divides. With logarithms, multiplications become addition problems (plus three look-ups in log and anti-log tables), and divisions become subtraction problems (again, plus three look-ups). What’s more, raising to powers and finding roots become reduced to multiplies and divides.
For example: find the 7th root of 12:
log 12 = 1.0792
Divide by 7 = 0.1542
Antilog of that = 1.426
There are other ways of finding 7th roots, but none so fast with pen and paper.
Pretty soon, someone realised that adding quantities was like putting marked rulers together: put 2 cm next to 3 cm and you get a total length of 5 cm, i.e., 5 is the sum of 2 and 3. So they invented rulers with logarithmic scales, called slide rules. And (again in the days of B.C.) scientists and engineers used slide rules to do their multiplies, divides and exponentiation. So the joke was that if you asked an engineeer, “What is 2 times 2?”, he would pull a slide rule out of his pocket and answer, “Approximately 4”
Anybody? I never saw the use for this stuff. I went through Calculus in high school, and in the intervening 30 years I have never used anything more than Trig even once. Some Algebra, some Trig, but that’s it.
1.) When I first learned them, I was taught that logs were originally invented by astronomers for their calculations. I don’t think it’s true – Napier and that bunch weren’t astronomers, but it’s probabl;y true that astronomers were the first big users.
2.) Physicists use them a lot. It’s hard to imagine developing, say, Statistical Mechanics without introdiucing logs and Stirling’s theorem and all.
3.) In the pre-computer and pre-calculator days most engineers used logs in some way, shape, or form. You used slide rulesw for calculating, log-log and semi-log paper for plotting and for seeing trends. I did my first two years of college without a calculator, using a slide rule to multiply and specialized paper to plot, in math, physics, chemistry, and materials science.
If b[sup]n[/sup] = x, then
[li]if you know b and n, and you want to know x, you use an exponential.[/li][li]if you know b and x, and you want to know n, you use a logarithm.[/li][li]if you know n and x, and you want to know b, you use a radical.[/li][/ul]
For example, in order to determine how much money you will have if you put $10,000 in an account with a compound interest of 5% per year, for 10 years, you would use an exponential: 10,000 * 1.05[sup]10[/sup] = 16288.95.
In order to determine how long it will take to double that initial $10,000, you need a logarithm: the base 1.05 log of 2 is 14.21, therefore 10,000 * 1.05[sup]14.21[/sup] = 20,000 (approximately), so it will take you a bit more than 14 years to double your money.
In order to determine how much interest you’d need to double your money in just 10 years, you use a radical: 2[sup]1/10[/sup] = 1.07177, hence you’ll need an account with 7.177% compound interest per year.
Corii mentioned a couple uses. Other examples are various logarithmic scales used in science: stellar luminosities, earthquake energies, and acidity (the pH factor).
In general, logs are often useful for quantities that either have a huge dynamic range (like the luminosities of stars), or undergo exponential growth or decay (like money earning compound interest, or a decaying radioactive element).
I use logs practically every day. Think about multiplying a bunch of probabilities together: .1*.2*.15*.07*… It becomes a really small number really fast. But take the log of it and it turns in to a sum: log(.1) + log(.2) + log(.15) + log(.07) + … Much more managable. The fact that logs preserve monotonicity and turn products into sums make them incredibly useful in statistics and computer science.
This sounds really silly, but I Really Got logs from playing a video game called Destiny of an Emperor.
In this game, you had five generals, each with an army, fighting the bad guys. I pretty quickly noticed something about the amount of damage they did- if you had 1000 men or 9000, you did about the same amount of damage. However, 999 men did a lot less damage than 1000. Each time you dropped below a power of 10, the amount of damage you did went way down.
I’m aware that this is a lousy model for damage done by an army, but it does illustrate logs pretty well- the damage you do is determined by (among some other things) the log of the number of men your general has. They probably used this model because there is a large dynamic range in the number of men in an army- you start off with about 100 men in each army, and end up with about 20,000 men in each army.