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. While the latter is obvious, the former may pose some problems – if you're not sure what the number e is, check out our e calculator. 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.
Symbolically, we can write the definition like so:įor comparison, the inverse operation that would return the 5 from 5⁸ would be simply the ( 8-th) root. 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 mathematicians 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.
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