PAIBOTLearn
Sign inSign up

Common logarithms

Exponents & logarithms

Japanese school year: Math II

What you learn

Learn how to use base-10 common logarithms to determine the number of digits and orders of magnitude for large powers. This application is crucial in scientific fields for estimating extremely large or small quantities like atomic particles or astronomical distances. Solid knowledge of basic logarithmic identities will ensure an easy transition to these calculations.

Go to practice

Key points

This relation uses the common logarithm log⁡10N\log_{10} N to determine the number of digits in a positive integer NN. If the integer part of log⁡10N\log_{10} N is n−1n - 1, then NN has nn digits.

10n−1≤N<10n  ⟺  n−1≤log⁡10N<n10^{n-1} \le N < 10^n \iff n - 1 \le \log_{10} N < n

This formula determines at which decimal place the first non-zero digit appears for a positive number N<1N < 1. It is determined by finding the integer nn satisfying −n≤log⁡10N<−(n−1)-n \le \log_{10} N < -(n - 1).

10−n≤N<10−(n−1)  ⟺  −n≤log⁡10N<−(n−1)10^{-n} \le N < 10^{-(n-1)} \iff -n \le \log_{10} N < -(n-1)

By rewriting 55 as 102\frac{10}{2}, this relation calculates log⁡105\log_{10} 5 using the given value of log⁡102\log_{10} 2. It is useful in problems where only log⁡102\log_{10} 2 is provided.

log⁡105=1−log⁡102\log_{10} 5 = 1 - \log_{10} 2

Choose a set to practice.