eli5: What exactly is a logarithm?
In: Mathematics
A logarithm is the exponent you need to make one number into the other number.
You already know 10^2 = 100 right? The 2 is the logarithm of 100 (for the base 10).
You know 2^3 = 8. The 3 is the logarithm of 8 for base 2.
Log is the exponent.
After we first came up with the idea of addition, we realized that there had to also be a ‘reverse’ operation, which would require us to ‘work backwards’ from an addition. An addition question would be “3 plus 5 equals what?” A subtraction question, then, could be expressed as “3 plus what equals 5?”
Later on we developed this operation called multiplication, which can be understood as repeated addition. “5 x 3”, means “5 + 5 + 5”. And this operation, multiplication, could have a ‘reverse’ version too! We call that division. A multiplication question is “3 times 5 equals what?”, and a division question is “3 times what equals 5?”
Now, exponentiation is repeated multiplication just like multiplication is repeated addition. “3^5” means “3x3x3x3x3” And a logarithm is *its* reverse operation. Log_3(5) is asking “3 to the what equals 5?”
It is just another function: log(x)
It takes an ‘x’ and it gives you a ‘y’
log(x) = y
It is defined as the inverse of the exponential function (some examples: 2^x, 77^x , e^x ).
What is an inverse function? It is easier to understand with examples:
Let’s say: 2^x = y
if x=3 then y = 2^3 = 8
Its inverse: log_2 (x) = y
does the opposite
if x = 8 then y = 3
——- Another example —–
if x = 5 then y = 2^5 = 32
Its inverse: log_2 (x) = y
does the opposite
if x = 32 then y = 5
There is addition. And when you add the same thing many times, it becomes quickly annoying:
2 + 2 + 2 + 2 = 8
So we have multiplication.
2 × 4 = 8
And we can determine how many additions were performed, using division:
3 × n = 27
n = 27 ÷ 3
—–
There is multiplication. And when you multiply the same thing many times, it becomes quickly annoying:
2 × 2 × 2 × 2 = 16
So we have exponentiation.
2⁴ = 16
And we can determine how many multiplications were performed, using *logarithms*:
3ⁿ = 27
n = log₃(27)
—–
Just like a division will not always give you whole numbers (7÷3 for instance), and it might seem absurd to perform an addition a non-integer number of times, logarithms will not always you give you whole numbers and it appears equally absurd.
Basically the opposite of an exponent.
If x^y =z, then logx(z)=y, ln() is a special case when x is equal to e, and if unspecified (log()) then x is assumed to be 10.
The logarithm of a number is easiest to think about as “how many digits do I have to write down to write this number?”. Think about the number 1 and the number 1000. 1000 is way bigger than 1 (a thousand times bigger), but it only takes four times as many digits to write down.
Because we use a number system where we have more than one digit (0-9), we can write big numbers without using a big number of digits. Imagine if we only had one number: 1. Then to write 5, we would have to write 11111. To write 10, we would have to write 1111111111. Writing 1000 would be absurd.
But because we can use more symbols, we can write big numbers in pretty small ways. In our case, we use 10 symbols (0-9) so we are in “base 10”. The logarithm of a number, base 10, is roughly the number of digits needed to write it down.
This isn’t exact – it’s only exactly true for numbers like 1, 10, 100, 1000, 10,000, and so forth. For the other numbers, the logarithm (base 10) is some number between the integers.
And we can use other bases. There’s binary (base 2), octal (base 8), and any other number of bases we could use. We can even use bases that aren’t integers. So as you go deeper the math will get a little more funky, but it’s always the same basic idea: how many digits does it take to write this number down using the place system?
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