When using the log function of awk, then what we get as a result is the natural logarithmic of the input, like:
...$ awk 'BEGIN{print log(100)}'
...$ 4.60517
So in order to obtain the logarithm of base 10 (or any other base), we just need to divide the result with the logarithm of the base, like:
...$ awk 'BEGIN{print log(100)}/log(10)'
...$ 2