What is the point of logarithms? How are they used? [closed]
You can "undo" addition by performing subtraction. You can "undo" multiplication by performing division.
When it comes to exponents, $x^y \not= y^x$, so you need two different "undo" functions.
Suppose that you know the value of $v$, and you know that this value was calculated by $v = x^n$.
If you know what $n$ was, then you can "take the $n$th root of $v$" to find $x$. That is, $x = \sqrt[n] v$.
If you know what $x$ was, then you can "take the base-$x$ logarithm of $v$" to find out what $n$ was. That is, $n = \log_x v$.
So that's what the logarithm function does. Why is that useful? Well, for the same reason that being able to undo an addition or a multiplication is useful. It lets you work backwards through a calculation. It lets you undo exponential effects.
Beyond just being an inverse operation, logarithms have a few specific properties that are quite useful in their own right:
Logarithms are a convenient way to express large numbers. (The base-10 logarithm of a number is roughly the number of digits in that number, for example.)
Slide rules work because adding and subtracting logarithms is equivalent to multiplication and division. (This benefit is slightly less important today.)
Lots of things "decay logarithmically". For example, hot objects cool down, cold objects warm up. Things in motion experience friction and drag and gradually slow down.
If you can take a problem and split it into two smaller problems that can be solved independently, you can probably write a computer program where the number of steps required to solve the problem is "logarithmic". That is, the time taken depends on the logarithm of the amount of data to be processed.
I'm sure there are lots of other examples.
Logarithms are defined as the solutions to exponential equations and so are practically useful in any situation where one needs to solve such equations (such as finding how long it will take for a population to double or for a bank balance to reach a given value with compound interest).
Historically, they were also useful because of the fact that the logarithm of a product is the sum of the logarithms and sums are easier to calculate by hand (or to estimate by overlapping rulers as in a slide rule). In addition to providing a computational "trick", this property is the basis of the mapping property described in Christian Blatter's answer and generalizes to the concept of self-adjoint generators of unitary groups, which has many mathematical applications and relates physical observables to symmetry properties in quantum mechanics.