Yes, they are closely related. In both cases it's about the base of the powers we use to describe a number. For number systems, saying we are using base ten means we express every number as a linear combination of integral powers of ten. For instance:
This should be well-known from elementary school, except, perhaps, the usage of exponents.
When we take logarithms base ten, we express a number as a single power of ten. For instance:
In both cases we see $10^2$, since $345$ is between $100$ and $1000$. But in the logarithm case, we take the $3$ in front of the $10^2$, and all the smaller terms, and "absorb" them into the exponent.
One difference between the two is that making sense of a base that is not an integer is easier for the logarithm than for the number system. We routinely do logarithms with base $e$, yet it's not entirely obvious that a base $e$ number system would even work.