### Logarithms in a nutshell

Someone asked me recently to make a post about logarithms. So here goes. I already answered the person in an email but I thought I could include some interesting history tidbits here, too.

Logarithms are simply the opposite operation of exponentation.

For example, from 23 = 8 we get log28 = 3, and we read it "base 2 logarithm of 8 equals 3".

So it's not difficult: if you understand how exponents work, logarithms have the same numbers, just in a little different places.

Just as in exponentiation, a logarithm has a base (2 in the above example). Remember that in 53, 5 is called the base and 3 is the exponent.

Other examples:
53 = 125 and log5125 = 3.
104 = 10,000 and log1010,000 = 4.
2x = 345 and log2345 = x.

As you can see from the last example above, you can use logarithms to solve equations where the x is the exponent:
4x = 1001
x = log41001.

Then you'd get the value of x from a calculator.

However, the base of the logarithm can be anything and most calculators only have two buttons: one for base 10 logarithm and another for base e logarithm (also called natural logarithm).

So you might have to convert the answer log41001 to one of those first.

And aren't we happy it CAN be done: log41001 = log101001/log104. So now you have two base 10 logs that you can punch into your calculator. And the answer is: x ≈ 4.983613129. Check: 44.983613129 = 1000.99999942, yeah, reasonably close I guess.

## History

Initially logarithms were used as calculation aids - before the age of calculators, of course. The way they work, people were able to avoid multiplying by using logarithms.

They would have a BIG book called the table of logarithms. And, if you had two really BIG numbers to multiply, you can imagine how tedious a task it is by hand. But with logarithms, let's say you had a and b to multiply as your numbers. You'd find their logarithms in the book, and simply ADD those logarithms - which is way easier to do by hand than multiplying: you'd find what log a + log b was.

Then, after you had the sum, you'd again check in the book "backwards" and find which number's logarithm was the sum you'd just calculated.

How does that work? If you're ready for more math and symbols, see below:

As you may remember, exponents have a property that:

bt bs = bs+t

Logarithms are the opposite of exponentiation... If you call A = bt and B = bs, then t = logbA and s = logbB.
Rewrite the original exponent property now using A and B:

AB = bs+t

And writing that in logarithm form you get:

logb AB = s + t = logbB + logbA.

So base b logarithm of a product (AB) is the same as the sum of the logarithms of the numbers. And this property allowed people to find products of large numbers by just adding the logarithms and checking 'backwards' which number had as its logarithm the obtained sum.

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Anonymous said…
Since log and exp are related in what is arguably the most intimate way two functions could be related with respect to their inner mechanics (i.e. inverse relationship), I don’t think any true understanding will be acquired until both are understood, with respect to the other.

My suggestion would be to teach log and exp as functions that give one information about a given tree diagram. Let’s say this tree exists out there in the Platonic realm, and log and exp are instruments that show us different aspects of it.

Exp is a function that says; given a branch factor b and the number of levels x, tell me how many leaf nodes = y are on the tree.

Log is a function that says; given a branch factor b and the number of leaf nodes y on the tree, tell me how many levels = x there are.

It then becomes clear that the number of levels x (returned by log()) grow arithmetically with respect to the branch factor b (think vertically), while the number of leaf nodes y (returned by exp()) grows geometrically with respect to the branch factor b (think horizontally).

Of course, this becomes less clean when you are dealing with reals instead of integers.
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Estefani Alarcón y Estefany López