What is the reason to introduce and study logarithmic functions? I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something that exponents already do just fine.

There's a lot of uses for logarithms
There are things you can solve without logarithms. Take for example:
${4.9}^x=34.185$
Without logarithms you might try taking the $x$'th root, and really just end up going in circles, with logarithms, we can solve this with logarithms this way.
$\log <!--\; \; -->{4.9}^x=\log <!--\; \; -->34.185$
$x\log <!--\; \; -->4.9=\log <!--\; \; -->34.185$
$x=\frac{\log <!--\; \; -->34.185}{\log <!--\; \; -->4.9}$

Logarithms were originally invented to make multiplication (as in, actually computing the product of two numbers by hand) easier. They were developed by one man, John Napier, in the 16th century, specifically as a method for doing by-hand multiplication.
The reasoning was as follows. It had already been noticed at the time that if you consider the sequence of powers of a number, say $1,2,4,8,\mathrm{16...}$, then multiplying any two numbers in that sequence can be done simply by adding together their positions in the sequence. This is simply another way of saying $x^a{x}^b=x^{a+b}$. Unfortunately, most numbers are not in the sequence $1,2,4,\mathrm{8...}$, so the trick isn't very useful.
Napier's original idea was to simply use a very small base, say $1.000001$. Then the sequence of its powers will grow very slowly, and so even though not every number is in the sequence, every number will at least be close to an element of the sequence. So if you need to multiply together two numbers, you can just approximate them by elements of the sequence and then use the trick discussed in the previous paragraph.
Using the concept of fractional exponents, which I'm sure you've seen if you're studying logarithms, we can do away with having to use a small base by effectively filling in the gaps between the elements of the sequence.
That's why logarithms were originally investigated. Nowadays we don't need to resort to tricks like that just to do multiplication. But once they had a name, they started showing up in all kinds of places. This is why we teach students about logarithms today. For example, in order to integrate $\frac{1}{x}$ in calculus, you "need the logarithm". Of course, you could just numerically integrate it, but it's useful to know that the result of that integration is actully a function with certain algebraic properties and which turns up as the answer to other problems as well.