Characteristic of Common Logarithm of Number Greater than 1

Theorem

Let $x \in \R_{>1}$ be a (strictly) positive real number greater than $1$.

The characteristic of its common logarithm $\log_{10} x$ is equal to one less than the number of digits to the left of the decimal point of $x$.

Proof

Let $x$ be expressed in scientific notation:

$x = a \times 10^e$

where:

$1 \le a < 10$

$e \in \Z_{\ge 0}$

From Range of Common Logarithm of Number between 1 and 10:

$0 \le \log_{10} a < 1$

The characteristic of $\log_{10} x$ equals $\map {\log_{10} } {10^e} = e$.

Thus the characteristic of $\log_{10} x$ is equal to the exponent of $x$.

By Multiplication by Power of 10 by Moving Decimal Point, multiplication by $10^e$ is the same as moving the decimal point $e$ places to the right.

Hence $a \times 10^e$ has $e$ more digits to the left of the decimal point than $a$ does.

That is: $e + 1$.

Hence the result.

$\blacksquare$

Sources

• 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Logarithms