Range of Common Logarithm of Number between 1 and 10

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Theorem

Let $x \in \R$ be a real number such that:

$1 \le x < 10$

Then:

$0 \le \log_{10} x \le 1$

where $\log_{10}$ denotes the common logarithm function.


Proof

We have:

\(\ds 1\) \(=\) \(\ds 10^0\) Definition of Integer Power
\(\ds 10\) \(=\) \(\ds 10^1\) Definition of Integer Power
\(\ds \leadsto \ \ \) \(\ds \log_{10} 1\) \(=\) \(\ds 0\)
\(\ds \log_{10} 10\) \(=\) \(\ds 1\)

The result follows from Logarithm is Strictly Increasing.


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Sources

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