It may seem that numbers always take a whole number of digits to write down: 139 and 931 are both written using three digits \('9', '3', and '1'\)\. However, as we will see, there's a sense in which 139 is barely using its third digit, while 931 is using all three of its digits to almost their full extent\. Logarithms quantify this intuition, and the precise answer that they give reveals some interesting information about how many digits it costs to write a given number down\.
This paragraph is good -- clearer than some of the other places where you tried to introduced this idea.