Common Logarithm of Number in Scientific Notation

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Theorem

Let $n$ be a positive real number which is presented (possibly approximated) in scientific notation as:

$n = a \times 10^d$

where:

$1 \le a < 10$
$d \in \Z$ is an integer.

Then:

$\log_{10} n = \log_{10} a + d$

where:

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


Proof

\(\ds n\) \(=\) \(\ds a \times 10^d\) by definition
\(\ds \leadsto \ \ \) \(\ds \log_{10} n\) \(=\) \(\ds \map {\log_{10} } {a \times 10^d}\)
\(\ds \) \(=\) \(\ds \log_{10} a + \log_{10} 10^d\) Logarithm of Product
\(\ds \) \(=\) \(\ds \log_{10} a + d\) Definition of Common Logarithm

We are given that:

$1 \le a < 10$

It follows from Range of Common Logarithm of Number between 1 and 10 that:

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

$\blacksquare$