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:
- $0 \le \log_{10} a < 1$
$\blacksquare$