Characteristic and Mantissa of a Common Logarithm

Characteristic and mantissa of common logarithm $\log_{10}x$. Since every real $x$ splits as $x=[x]+\{x\}$ (integer part plus fractional part), the same split applies to its logarithm: $\log_{10}x=[\log_{10}x]+\{\log_{10}x\}$, where the integral part $[\log_{10}x]$ is called the characteristic, and the fractional part $\{\log_{10}x\}$ is called the mantissa.

Illustration: for $\log_{10}23$: $\log_{10}10<\log_{10}23<\log_{10}100$, i.e. $\log_{10}10<\log_{10}23<\log_{10}10^2$, i.e. $\log_{10}10<\log_{10}23<2\log_{10}10$, i.e. $1<\log_{10}23<2$ (since $\log_{10}10=1$). So $[\log_{10}23]=1$; hence the characteristic of $\log_{10}23$ is $1$.

In general: the characteristic of the logarithm of a number $N$, with $m$ digits in its integral part, is $m-1$.

Ex. 16: Given $\log_{10}2=0.3010$, find the number of digits in the number $20^{10}$. …