We have to remember that there are also fractional exponents, and indeed, here, we need one of those. To be precise, it's not an integer power of 2. Similarly, log₂ 16 = 4 or log₂ 32 = 5.īut what is, say, log₂ 5 ? Surely, 5 is not a power of 2. For instance, we can say that the log with base 2 of 8 is 3. Now we can see some more examples than just the log₂ 4 = 2 from above. Remember that the exponent can also be 0 or even negative. Since it's so important, let's recall a few basic powers of 2. Moreover, it's the base for any computer-related operations via the binary representation. E.g., it is the smallest prime number and the only even one. However, it has some interesting properties. For instance, we can easily observe that log₂ 4 = 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.

Well, on second thought, why don't we dedicate a whole section to this one?Īs mentioned at the end of the above section, the binary logarithm is a special case of the logarithmic function with base 2. In essence, we'll focus on taking the powers of 2 and. Today, we'll focus on a very special case of the logarithm, i.e., base 2, which we sometimes call the binary logarithm. Also, quite a few physical units are based on logarithms, for instance, the Richter scale, the pH scale, and the dB scale. 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). After all, whatever we raise to power 0, we get 1. Whatever the base, the logarithm of 1 is equal to 0.In other words, whenever we write logₐ(b), we require b to be positive. The logarithm function is defined only for positive numbers.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. There are two very special cases of the logarithm which have unique notation: the natural logarithm and the logarithm with base 10.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.īefore we move further, let us have a pretty bullet list with a few vital points of information about our new friend, the logarithm function. 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. Note, however, that in general, this can be a fractional exponent.įor comparison, the inverse operation that would return the 5 from 5⁸ would be simply the ( 8-th) root.

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. In other words, it is a function that tells you the exponent needed to obtain the value. The logarithm (of base 5) would be the operation if we chose option 8. So what should the inverse operation give? If we have 5⁸, should it return 5 or 8? After all, we know that 5 + 8 = 8 + 5 and 5 * 8 = 8 * 5, but 5⁸ is very different from 8⁵. For exponents, however, the story gets more complicated. For multiplication, it's still pretty simple: it's division. Lucky for us, mathematics, and the whole world of science, other curious people found the answer.įor addition, it was easy: the inverse operation is subtraction. In this case, they wondered if there was a way to invert all these operations. However, there is always that one curious person who asks the wildest questions. 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. As soon as humanity learned to add numbers, it found a way to simplify the notation for adding the same number several times: multiplication.

Logarithmic condense calculator