Welcome to Omni's log base 2 calculator. Your favorite tool to calculate the value of
log₂(x) for arbitrary (positive)
x. The operation is a special case of the logarithm, i.e. when the log's base is equal to
2. As such, we sometimes called it the binary logarithm.
So what is, e.g., the log with base
log₂ 16? Or
log₂ 32? Well, let's jump straight into the article and find out!
What is a logarithm?
5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 = 8 * 5
Then, an obvious question appeared: how could we write multiplying the same number several times? And again, there came some smart mathematician who introduced exponents.
5 * 5 * 5 * 5 * 5 * 5 * 5 * 5 = 5⁸
However, there is always that one curious person who asks the wildest questions. In this case, they wondered if there was a way to invert all these operations. Lucky for us, mathematics, and the whole world of science, other curious people found the answer.
For addition, it was easy: the inverse operation is subtraction. For multiplication, it's still pretty simple: it's division. For exponents, however, the story gets more complicated. After all, we know that
5 + 8 = 8 + 5 and
5 * 8 = 8 * 5, but
5⁸ is very different from
8⁵. So what should the inverse operation give? If we have
5⁸, should it return
The logarithm (of base
5) would be the operation if we chose option
8. In other words, it is a function that tells you the exponent needed to obtain the value. Symbolically, we can write the definition like so:
For comparison, the inverse operation that would return the
5⁸ would be simply the (
8-th) root. If we wanted to get a bit more technical, then we could say that, in general, if we had an expression
xʸ, then the root is the inverse operation for
x, while the logarithm is that for
y. And if we wanted to get even more technical, we'd say that the first inverts a polynomial function, while the latter inverts an exponential one.
Before we move further, let us have a pretty bullet list with a few vital points of information about our new friend, the logarithm function.
- There are two very special cases of the logarithm which have unique notation: the natural logarithm and the logarithm with base
10. We denote them
log(x)(the second one simply without the small
10), and their bases are, respectively, the Euler number
eand (surprise, surprise!) the number
- The logarithm function is defined only for positive numbers. In other words, whenever we write
logₐ(b), we require
bto be positive.
- Whatever the base, the logarithm of
1is equal to
0. After all, whatever we raise to power
0, we get
- Logarithms are extremely important. And we mean EXTREMELY important. Outside of mathematics, they're used in statistics (e.g., the lognormal distribution), economy (e.g., the GDP index), medicine (e.g., the QUICKI index), and chemistry (e.g., the half-life decay). Also, quite a few physical units are based on logarithms, for instance, the Richter scale, the pH scale, and the dB scale.
Today, we'll focus on a very special case of the logarithm, i.e., base
2, which we sometimes call the binary logarithm. In essence, we'll focus on taking the powers of
2 and... Well, on second thought, why don't we dedicate a whole section to this one?
As mentioned at the end of the above section, the binary logarithm is a special case of the logarithmic function with base
2. That means that we'll have expressions of the form
log₂(x), and we'll ask ourselves to what power we should raise
2 in order to obtain
x. For instance, we can easily observe that
log₂ 4 = 2.
2 is a number like any other. However, it has some interesting properties. E.g., it is the smallest prime number and the only even one. Moreover, it's the base for any computer-related operations via the binary representation.
Since it's so important, let's recall a few basic powers of
2. Remember that the exponent can also be
0 or even negative.
Now we can see some more examples than just the
log₂ 4 = 2 from above. For instance, we can say that the log with base
log₂ 16 = 3 or
log₂ 32 = 5.
But what is, say,
log₂ 5? Surely,
5 is not a power of
To be precise, it's not an integer power of
2. We have to remember that there are also fractional exponents, and indeed, here, we need one of those. Unfortunately, they're not so simple to guess. In some cases, we can try to use tricks like the change of base formula, but, in general, it's best to use an external tool - something like our log base 2 calculator.
In it, you can see two variable windows:
log₂(x). Hopefully, the notation is self-explanatory. For example, if you'd like to find
log₂ 16, you need to input
x, and the calculator will give you the answer in the other window. If you require
log₂ 32, you enter
32. Also, note how Omni's log base 2 calculator works both ways: you can either input the value of
x and obtain
log₂(x) or the other way round.
That'll be enough for today's lesson. Go, my young padawan, and make sure to play around with the calculator or any other algebra-related tool that we have on offer.