What is a real world example of a logarithm?
A basic concept in Information Theory is the Entropy os a random variable. Given a random variable X with alphabet $\displaystyle \chi= \{1,2,...,m\}$ , the the Entropy of X is ...
$\displaystyle H(X)= - \sum_{x \in \chi} p(x)\ \ln p(x)$ (1)
... where $\displaystyle p(x)=P\{X=x\}$...
Kind regards
$\displaystyle \chi$ $\displaystyle \sigma$
Google is your friend. As it turns out, Wikipedia is packed full of valid real-world applications of logarithms. See this link.