Question

What will be the value of the expression?
$\sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} $

Answer 1

Hint – In this question we have to find the value of the given expression so simply use the basic logarithm property ${\log _b}a = \dfrac{{\log a}}{{\log b}}$, along with other logarithm properties to get the right answer.

Complete step by step answer:
$\sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} $
As we know that ${\log _b}a = \dfrac{{\log a}}{{\log b}}$ so use this property in above equation we have,
$ \Rightarrow \sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} = \sqrt {{{\left( {\dfrac{{\log 4}}{{\log 0.5}}} \right)}^2}} = \sqrt {{{\left( {\dfrac{{\log {2^2}}}{{\log \dfrac{1}{2}}}} \right)}^2}} $
And we also know that $\log {a^b} = b\log a,{\text{ }}\log \dfrac{a}{b} = \log a - \log b$ and the value of $\log 1 = 0$ so use these properties simplify the above equation we have,
$ \Rightarrow \sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} = \sqrt {{{\left( {\dfrac{{\log {2^2}}}{{\log \dfrac{1}{2}}}} \right)}^2}} = \sqrt {{{\left( {\dfrac{{2\log 2}}{{\log 1 - \log 2}}} \right)}^2}} = \sqrt {{{\left( {\dfrac{{2\log 2}}{{0 - \log 2}}} \right)}^2}} $
Now as we see log2 is cancel out from numerator and denominator so we have,
$ \Rightarrow \sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} = \sqrt {{{\left( { - 2} \right)}^2}} $
Now as we know square of any negative value is positive so we have,
$ \Rightarrow \sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} = \sqrt 4 = \sqrt {{2^2}} $
Now as we know the square root of any positive number gives us a $ \pm $ number.
$ \Rightarrow \sqrt {{{\left( {{{\log }_{0.5}}4} \right)}^2}} = \sqrt {{2^2}} = \pm 2$
So, this is the required answer.

Note – Whenever we face such types of problems the key concept is to have the good gist of the logarithmic identities as well as basic properties. Some of logarithmic important identities are being mentioned above. Remember while dealing with square root the domain of square root functions need to be taken care of as it is defined only for values greater than equal to 0.