Question

How do you simplify ${\log _{10}}\left( {{{10}^{\dfrac{1}{2}}}} \right)$?

Answer 1

Hint: The given problem deals with the use of logarithms. It focuses on the basic definition of logarithm function and its properties. For such types of questions that require us to simplify logarithmic expressions, we need to have knowledge of all the properties of logarithm function and applications of each one of them.
Formula used:
Properties of Logarithm function: Following are some useful properties of logarithm function:
1. ${\log _a}\left( {\dfrac{x}{y}} \right) = {\log _a}\left| x \right| - {\log _a}\left| y \right|$, where $a > 0,a \ne 1$ and $\dfrac{x}{y} > 0$
2. ${\log _a}\left( {{x^n}} \right) = n\log \left| x \right|$, where $a > 0,a \ne 1$ and ${x^n} > 0$
3. ${\log _a}\left( a \right) = 1$, where $a > 0,a \ne 1$

Complete step by step solution:
In the given problem, we are required to simplify ${\log _{10}}\left( {{{10}^{\dfrac{1}{2}}}} \right)$. This simplification can be done with the help of logarithm properties.
So, there are various logarithm properties that can be used to simplify the given algorithmic expression ${\log _{10}}\left( {{{10}^{\dfrac{1}{2}}}} \right)$.
Using the logarithm property ${\log _a}\left( {{x^n}} \right) = n\log \left| x \right|$, where $a > 0,a \ne 1$ and ${x^n} > 0$, we get
$ \Rightarrow \dfrac{1}{2}{\log _{10}}\left( {10} \right)$
Now, by basic definition of logarithm function and understanding of interconversion of logarithm function to exponential function, we know that ${\log _{10}}\left( {10} \right) = 1$.
$ \Rightarrow \dfrac{1}{2}$

Hence, ${\log _{10}}\left( {{{10}^{\dfrac{1}{2}}}} \right)$ can be simplified as $\dfrac{1}{2}$ by the use of logarithm properties and identities.

Note: The given problem involves use of properties and identities of log function and hence requires us to have a thorough knowledge of the same. We also need to have a basic idea about the applications of the identities and properties in such questions.