common logarithm
Because we count in tens, the logarithm with base 10 is so handy that it earned the name common logarithm. Its question is intuitive: roughly how many digits does a number have, or what power of 10 sits closest to it? The common log of 1000 is 3 because 1000 is 10 to the third power.
We write it as log(x) — when a logarithm appears with no base shown in a calculator or a textbook of science or engineering, base 10 is almost always meant. Formally log(x) = log_10(x), so log(x) = y is the same statement as 10^y = x. Each step of 1 in the common log corresponds to multiplying x by 10.
This base-10 view is the backbone of scientific notation and of many measurement scales. A number around 10^6 has a common log near 6; a number around 10^(-3) has a common log near -3. So the integer part of a common log essentially tells you the order of magnitude — how many powers of ten the number carries.
log(10000) = 4 since 10^4 = 10000. And log(50) ≈ 1.699, which sits between 1 and 2 because 50 lies between 10^1 = 10 and 10^2 = 100.
The integer part of a common log reveals the number's order of magnitude.
Watch out: pure-mathematics texts (and some software) sometimes write “log” to mean the natural log, base e. Always confirm which base an unlabeled “log” means in a given source.