Question

How do you simplify ${\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right)$?

Answer 1

A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.
$y = {b^x}$exponential form,
$x = {\log _b}y$ logarithmic function, where $x$ is the logarithm of $y$ to the base $b$, and ${\log _b}y$ is the power to which we have to raise $b$ to get $y$, we are expressing $x$ in terms of $y$.
Now the given question can solved by using properties of logarithms i.e.,$\log x - \log y = \log \left( {\dfrac{x}{y}} \right)$,and ${\log _a}a = 1$ solve the expression to get the required result.

Complete step-by-step answer:
We know that logarithm is the power to which a number must be raised in order to get some other number, and the base unit is the number being raised to a power, For example, the base ten logarithm of 1000 is 3, because ten raised to the power of two is 100 , because${10^3} = 1000$. In general, you write log followed by the base number as a subscript. The most common logarithms are base 10 logarithms and natural logarithms; they have special notations. A base ten log is written as$\log$, and we use different base unit but most common logarithms are base 10 logarithms.
Now given equation is${\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right)$,
Now using logarithmic property$\log x - \log y = \log \left( {\dfrac{x}{y}} \right)$, then the equation becomes,
$\Rightarrow {\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right) = {\log _{10}}\left( {\dfrac{{\dfrac{1}{{100}}}}{{\dfrac{1}{{1000}}}}} \right)$,
Now simplifying the right hand side, we get,
$\Rightarrow {\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right) = {\log _{10}}\left( {\dfrac{1}{{\left( {\dfrac{1}{{10}}} \right)}}} \right)$,
Now taking the denominator of the denominator to the numerator we get,
$\Rightarrow {\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right) = {\log _{10}}\left( {10} \right)$,
By using identity${\log _a}a = 1$, we get,
$\Rightarrow {\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right) = 1$.
The value of${\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right)$is equal to$1$.
Final Answer:

$\therefore$ The value of${\log _{10}}\left( {\dfrac{1}{{100}}} \right) - {\log _{10}}\left( {\dfrac{1}{{1000}}} \right)$ is equal to $1$.

Note:
A logarithm is a mathematical operation that determines how many times a certain number, called the base, is multiplied by itself to reach another number, in these types of questions, we use logarithmic properties and formulas, and some of useful formulas are:
${\log _a}xy = {\log _a}x + {\log _a}y$,
$\log x - \log y = \log \left( {\dfrac{x}{y}} \right)$
${\log _a}{x^n} = n{\log _a}x$,
${\log _a}b = \dfrac{{{{\log }_e}b}}{{{{\log }_e}a}}$,
${\log _{\dfrac{1}{a}}}b = - {\log _a}b$,
${\log _a}a = 1$,
${\log _{{a^x}}}b = \dfrac{1}{x}{\log _a}b$