Take the 1st order Taylor series the function $log(x)$. around one:
$$ log(x) \approx log(1) + \frac{x-1}{1} = x-1$$
Therefore, for a random variable r that is close to zero:
$$ log(1+r) \approx r$$
Log returns are typically something like $log(P_{new}/P_{old})$:
$$\Delta log(P_t) = log(P_{new}/P_{old}) = log((P_{old} + \Delta P)/P_{old}) = log(1 + \Delta P/P_{old}) \approx \Delta P/P_{old}$$
But $\Delta P/P_{old}$ is just the percent change in $P_{old}$. We've therefore shown that we can approximate the percent change in a variable $X$ with $\Delta log(X)$.
Why bother? Mostly because log differences can be summed but percent changes cannot be:
Using logs, or summarizing changes in terms of continuous compounding,
has a number of advantages over looking at simple percent changes. For
example, if your portfolio goes up by 50% (say from \$100 to \$150) and
then declines by 50% (say from \$150 to \$75), you’re not back where you
started. If you calculate your average percentage return (in this
case, 0%), that’s not a particularly useful summary of the fact that
you actually ended up 25% below where you started. By contrast, if
your portfolio goes up in logarithmic terms by 0.5, and then falls in
logarithmic terms by 0.5, you are exactly back where you started. The
average log return on your portfolio is exactly the same number as the
change in log price between the time you bought it and the time you
sold it, divided by the number of years that you held it.
Jame Hamilton: Use of logarithms in economics