significance of logarithms in computing large numbers

Question

I'm considering the use of various strategies that allow very large numbers to be computed rather than simply giving errors. One way apparently is to use logarithms. Taking the log of every term in an expression allows it to be worked on by it's exponent number (along with the base) so that the very large result can be avoided until the very final answer. I'm still shaky on why this happens (logically) and how the same final answer can be gotten since the logs have to be computed anyhow. So to me it would seem that this simply delays the log calculation. IOW, I'm not seeing how the very large terms are avoided.
Second, please share any other mathematical tactics that can be used to allow very large numbers to be computed. Thank you.