Question

What is the value of the common logarithm $\log \left( {10,000} \right)$ $?$

Answer 1

Hint : First we have to know the common $\log$ is the Logarithms in base $10$ is the power of $10$ that produces that number. First the given number is expressed as the $n$ power of $10$ . Then $n$ is the value of the common logarithm of that number.

Complete step-by-step answer :
Suppose $p$ and $q$ are any two non-zero positive real numbers the following formulas holds:
${\log _n}\left( {pq} \right) = {\log _n}\left( p \right) + {\log _n}\left( q \right)$ .
${\log _n}\left( {\dfrac{p}{q}} \right) = {\log _n}\left( p \right) - {\log _n}\left( q \right)$ .
${\log _n}\left( {{p^a}} \right) = a{\log _n}\left( p \right)$ .
${\log _p}\left( q \right) = \dfrac{{{{\log }_n}\left( q \right)}}{{{{\log }_n}\left( p \right)}}$ .

Given ${\log _{10}}\left( {10000} \right)$ ---(1)
Since we know that $10000 = {10^4}$ , then (1) becomes
${\log _{10}}\left( {{{10}^4}} \right)$
Using the third formula we get
${\log _{10}}\left( {{{10}^4}} \right) = 4{\log _{10}}\left( {10} \right)$
Using the fourth formula, we get
$4{\log _{10}}\left( {10} \right) = 4 \times \dfrac{{{{\log }_n}10}}{{{{\log }_n}10}} = 4$
Hence ${\log _{10}}\left( {10000} \right) = 4$ .
The domain of the common log as well as the logarithm in any base, is $x > 0$ . We cannot take a $\log$ of a negative number, since any positive base cannot produce a negative number, no matter what the power.
So, the correct answer is “4”.

Note : Note that the most common error is forgetting that the function does not exist for values of $x \leqslant 0$ . The result of the common $\log$ function is simply the variable $y$ for the equation $x = {10^y}$ . As there is no value for $y$ (in the domain of real numbers) for which $x \leqslant 0$ , the domain for the inverse function (our common log) is $0 < x < \infty$ . Also note that if ${\log _n}y$ is greater than ${\log _n}x$ . that means that $y$ is greater than $x$ by a factor of $n$ .