Introduction
I had a conversation with a high school student this week about logarithms and the cost of college tuition. She asked her math teacher for some uses of logarithms, a question he could not answer. That's a shame, since they are so useful. One way that everyone uses them without being consciously aware of it is when dealing with large sums of money.
Think of millionaires. They have a lot of money. When you think of billionaires, how do you think of them? I think "they are one step beyond millionaires." And if there were trillionaires, they'd be one step beyond billionaires. You may also think "a trillion has 3 more zeros than a billion, which has 3 more zeros than a million." But this hides the real scale. A single billionaire has as much money as a town full of millionaires, and a trillion dollars is unfathomably large. However, counting the "number of steps" in an exponentially growing value (like the number of decimal places in a number) is a logarithm.
The Question
But what if we stopped thinking of large numbers in terms of logarithms? How large is a trillion dollars?
Let's start small. Most people believe that a college education is expensive, that it's a lot of money, that it is expensive to send your kid to college. Let's see if that's true.
The Math
And now for the war. It doesn't matter if you are for the American wars or against them, the math is agnostic. And we can compare war costs with college tuition. I read recently that the US cost of the Iraqi and Afghan wars total 4.4 trillion dollars (http://costsofwar.org/article/economic-cost-summary).
Here are the statistics:
- 3.3 million US high school grads per year (http://nces.ed.gov/fastfacts/display.asp?id=372)
- $4.4 trillion: US cost of Iraqi and Afghan wars (http://costsofwar.org/article/economic-cost-summary)
- $10 thousand / year: average in-state tuition for US public colleges (https://bigfuture.collegeboard.org/pay-for-college/college-costs/college-costs-faqs#)
- 3.25% prime interest rate (http://data.worldbank.org/indicator/FR.INR.LEND)
- 4 years: length of college degree
The US spends $4.4 trillion * 3.25% = $143 billion per year on interest for the previous two wars.
And $143 billion per year / $10 thousand tuition = 14 million college tuitions per year.
And 14 million per year / 4 years = 3.3 million four year college tuitions.
And there are 3.3 million high school graduates per year.
That is, we can pay the tuition cost of a four year in-state college degree for every high school graduate in the US (for the rest of time!) with the interest on our war debt (assuming a stable population, interest rate, and tuition cost, and that the US does not pay down its national debt).
Note that tuition cost might not rise like it does now if it were prepaid by the government. This can be verified by looking at the cost of public school in the US or the cost of post-secondary school in countries that pay for every child to get a college education.
And said another way, because 1 / 3.25% = 30, the US citizens, as a whole, believe that it is 30 times more important to fight people in Iraq and Afghanistan than giving every child in the country a college degree.
Addendum
We use logarithms every time we measure an exponential function in terms of steps. Here are some examples:
- Number of digits in a number.
- Scientific notation.
- Number of years before the word population hits 8 billion.
- Number of years before the rain forests are cut down.
- Number of years before your investment reaches $100,000.
- Half-life of a radioactive isotope.
- How loud your stereo is (in Decibels).
- How strong an earthquake is (in Richters).
- Notes on the musical scale (frequency increases exponentially).
- The height of a binary search tree.
- Whatever you're talking about when you say "order of magnitude".
