What are logarithms used for?
- earthquake magnitudes,
- noise levels in decibels, and
- acidity (pH).
A big earthquake can be millions of times bigger than a tiny one. If you tried to make a bar graph where the bars has sizes 10, 100, and 10 000 000, it would look stupid. The bars of size 10 and 100 would be too small to see, and you won’t be able to tell that one of them is ten times bigger than the other. If you instead take the logarithm of each number, you get 1, 2, and 7. That makes a bar graph you can understand.
Keep that in mind when you hear about earthquake magnitudes. A 7.0 earthquake is ten times bigger than a 6.0 earthquake, which is ten times bigger than a 5.0 earthquake. Taking logarithms lets us put an earthquake caused by a stick of dynamite (1.2) on the same scale as the 2011 earthquake in Japan (9.0).
Logarithms can also be used to measure how long it will take something to grow exponentially or decay exponentially, such as
- money growing with a fixed interest rate,
- bacteria growing in a petri dish,
- radioactive decay, for example in , and
- the sound made by a bell.
If you have bacteria that divide every 30 minutes and are currently taking up 0.1% of the petri dish, you can use logarithms to estimate how long it will take them to fill up the entire dish. The same goes for $5000 in an account with a 2% interest rate. If you leave the interest in the account, logarithms will tell you when you’ll have $6000.
Logarithms can also be used in calculations by
- turning multiplication into addition.
If I gave you the option between multiplying twenty numbers together by hand or adding twenty numbers together by hand, you’d pick the second option. If you need to multiply twenty numbers, you can instead take the base 10 logarithm of each number, add the results, and then raise 10 to that power. Finding the logarithm might seem hard, but, in the past, people could just look it up in a logarithm table or use a slide rule. Finding the answer using logarithms was way faster.
The problem of multiplying lots of numbers was the original reason logarithms were developed. This method is now obsolete thanks to computers, which are pretty fast at multiplying. In the meantime, though, we’ve discovered tons of uses of logarithms, most of which I haven’t even listed here.