# Log Base 2 Calculator

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 `2`

of `8`

? Or `log₂ 16`

? Or `log₂ 32`

? Well, let's jump straight into the article and find out!

## What is a logarithm?

As soon as humanity learned to add numbers, it found a way to simplify the notation for **adding the same number** several times: multiplication.

`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 `5`

or `8`

?

**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:

💡 `logₐ(b)` gives you the power to which you'd need to raise `a` in order to obtain `b` . Note, however, that in general, this can be a fractional exponent. |

For comparison, the inverse operation that would return the `5`

from `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`ln(x)`

and`log(x)`

(the second one simply without the small`10`

), and their bases are, respectively, the Euler number`e`

and (surprise, surprise!) the number`10`

. **The logarithm function is defined only for positive numbers.**In other words, whenever we write`logₐ(b)`

, we require`b`

to be positive.- Whatever the base,
**the logarithm of**`1`

**is equal to**`0`

. After all, whatever we raise to power`0`

, we get`1`

. **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?

## Binary logarithm

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`

.

Seemingly, `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.

x | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

2^{x} |
⅛ | ¼ | ½ | 1 | 2 | 4 | 8 | 16 | 32 | 64 | 128 | 256 |

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 `2`

of `8`

is `3`

. Similarly, `log₂ 16 = 3`

or `log₂ 32 = 5`

.

**But what is, say,** `log₂ 5`

**?** Surely, `5`

is not a power of `2`

.

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**: `x`

and `log₂(x)`

. Hopefully, the notation is self-explanatory. For example, if you'd like to find `log₂ 16`

, you need to input `16`

under `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.