A basic practical reason to care about logarithms is that there are many numbers in real life which vary greatly in size, and it is both a pain and misleading to compare their sizes directly; one should instead compare the sizes of their *logarithms*, for various reasons. This is why the Richter scale is logarithmic; see these Wikipedia articles for some examples.

Logarithms also appear in the basic mathematical description of information. Suppose I send you a message consisting of zeroes and ones. If the message has length $n$, we say that it contains $n$ *bits* of information. There are $2^n$ possible such messages, which leads to a general principle: whenever you are in a situation where there are $k$ possibilities and you know that one of them happens, you have gained $\log_2 k$ bits of information.

Information is a fundamental concept. Consider the following puzzle: you have $1000$ bottles of wine, and you know that one of them is poisoned. You have an indeterminate number of rats to which you can feed various wines; if they are poisoned, they will die in $1$ hour. How many rats do you need to figure out which bottle is poisoned in $1$ hour?

The answer is $10$. This is because you want to figure out which of $1000$ possibilities happens, so you want to gain $\log_2 1000 \approx 10$ bits of information. If you feed $n$ rats some amount of wine, the amount of information you have after $1$ hour is precisely a list of which rats died and which rats didn't - zeroes and ones - so you have gained at most $n$ bits of information. (You might not reach this upper bound if some of the information you gain is redundant.) This requires that $n$ is at least $10$, and in fact this bound can be achieved by the following algorithm:

Label the wines $0, 1, 2, ... 999$ and convert the numbers to binary. Each of these numbers has at most $10$ binary digits. Assign each of the rats wines as follows: rat $i$ will drink all the wines with the property that the $i^{th}$ binary digit is $1$. After $1$ hour, the pattern of which rats die spells out the binary expansion of the poisoned wine.

I really like this problem because the problem statement does not mention logarithms at all, but it is an inevitable consequence of the particular optimization you are trying to accomplish that logarithms appear in the solution.