How do you simplify \[\log 10\left( \sqrt{10} \right)\]?

Hint: A common logarithm is the logarithm of base 10. To get the logarithm of a number n, find the number x that when the base is raised to that power, the resulting value is n. The few properties of logarithm required for evaluating are \[\log ab=\log a+\log b\], \[{{\log }_{b}}{{a}^{p}}=p{{\log }_{b}}a\]. If bases are equal, then we can use \[{{\log }_{b}}{{b}^{a}}=a\].

Complete step by step answer:
As per the given question, we have to evaluate the given logarithmic expression. We can easily simplify the given logarithmic expression using the logarithmic and exponents properties. Here, we have the logarithmic expression \[\log 10\left( \sqrt{10} \right)\].

In the given expression, let \[a=10\] and \[b=\sqrt{10}\]. Then the expression looks like
\[\Rightarrow \log 10(\sqrt{10})=\log ab\]
Using the property \[\log ab=\log a+\log b\], we can write \[\log 10\left( \sqrt{10} \right)\] as the sum of \[\log 10\] and \[\log \sqrt{10}\]. That is, we can express it as
\[\Rightarrow \log 10\left( \sqrt{10} \right)=\log 10+\log \sqrt{10}\] ----(1)
As we know that \[\sqrt{10}\] is nothing but \[{{10}^{\dfrac{1}{2}}}\] and 10 is \[{{10}^{1}}\], we can rewrite the equation (1) as
\[\Rightarrow \log 10\left( \sqrt{10} \right)=\log {{10}^{1}}+\log {{10}^{\dfrac{1}{2}}}\] ----(2)

In the equation (2), we can use the property \[{{\log }_{b}}{{b}^{a}}=a\] as the base is 10 and the number are 10 raised to certain powers. Using the property \[{{\log }_{b}}{{b}^{a}}=a\], we get \[\log {{10}^{1}}=1\] and \[\log {{10}^{\dfrac{1}{2}}}=\dfrac{1}{2}\]. So, we can substitute these values into the equation (2) to get the modified expression given below
\[\Rightarrow \log 10\left( \sqrt{10} \right)=1+\dfrac{1}{2}\]
On adding 1 and \[\dfrac{1}{2}\], we get \[\dfrac{3}{2}\]. So, the expression can be written as \[\Rightarrow \log 10\left( \sqrt{10} \right)=\dfrac{3}{2}\]

\[\therefore \dfrac{3}{2}\] is the simplified form of the given logarithmic expression \[\log 10(\sqrt{10})\].

Note: We can find the value of \[\log 10(\sqrt{10})\] directly by writing \[\sqrt{10}\] as \[{{10}^{1/2}}\] and adding the powers to get \[{{10}^{1+\dfrac{1}{2}}}={{10}^{\dfrac{3}{2}}}\]. Then, using the property \[{{\log }_{b}}{{b}^{a}}=a\], we get \[\dfrac{3}{2}\] as the answer. We should remember that we are dealing with a logarithmic function which doesn’t exist for \[x\le 0\]. We should avoid calculation mistakes to get the desired results.