Question

The value of the expression \[\sqrt {\log _{0.5}^24} \]is

Answer 1

Hint: We start by trying to simplify everything with powers of 2. Then we use the general log properties \[{\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{{{\text{logb}}}}{{{\text{loga}}}}\] and \[{\text{log}}{{\text{b}}^{\text{a}}}{\text{ = alogb}}\] to proceed. And we find the answer by simple mathematical processes.

Complete step by step Answer:

Given:\[\sqrt {\log _{0.5}^24} \],
We know that \[\sqrt {\log _{0.5}^24} \] can be written as \[\sqrt {(\log _{0.5}^{}4} {)^2}\]
Now,
We also have, \[{\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{{{\text{logb}}}}{{{\text{loga}}}}\]
We are going to use this property here,
So, \[{\text{lo}}{{\text{g}}_{0.5}}{\text{4 = }}\dfrac{{{\text{log4}}}}{{{\text{log0}}{\text{.5}}}}\]
Now, \[\sqrt {(\log _{0.5}^{}4} {)^2}\]\[ = \sqrt {{{\left[ {\dfrac{{{\text{log4}}}}{{{\text{log0}}{\text{.5}}}}} \right]}^2}} \]
we know that $0.5 = \dfrac{1}{2}$ and $\dfrac{1}{2} = {2^{ - 1}}$
\[ = \sqrt {{{\left[ {\dfrac{{{\text{log}}{{\text{2}}^2}}}{{{\text{log}}{{\text{2}}^{ - 1}}}}} \right]}^2}} \]
We know that \[{\text{log}}{{\text{b}}^{\text{a}}}{\text{ = alogb}}\]
\[ = \sqrt {{{\left[ {\dfrac{{{\text{2log2}}}}{{{\text{ - log2}}}}} \right]}^2}} \]
On simplification we get,
\[ = \sqrt {{{( - 2)}^2}} \]
\[ = \sqrt 4 = 2\]
So, we have the value of \[\sqrt {\log _{0.5}^24} \]is 2.

Note: We have used some general logarithmic formulas here like,\[{\text{lo}}{{\text{g}}_{\text{a}}}{\text{b = }}\dfrac{{{\text{logb}}}}{{{\text{loga}}}}\],\[{\text{log(}}\dfrac{{\text{a}}}{{\text{b}}}{\text{) = - log(}}\dfrac{{\text{b}}}{{\text{a}}})\],\[{\text{log}}{{\text{b}}^{\text{a}}}{\text{ = alogb}}\] these are the ones you need to remember.
Let us prove one of the properties i.e.\[{\text{log(}}\dfrac{{\text{a}}}{{\text{b}}}{\text{) = - log(}}\dfrac{{\text{b}}}{{\text{a}}})\]
On solving the left-hand side
\[ \Rightarrow {\text{log}}\left( {\dfrac{{\text{a}}}{{\text{b}}}} \right){\text{ = log}}{\left( {\dfrac{b}{a}} \right)^{ - 1}}\]
Using \[ \Rightarrow {\text{log}}\left( {{{\text{x}}^a}} \right){\text{ = alog}}\left( x \right)\]
\[{\text{ = - log}}{\left( {\dfrac{b}{a}} \right)^{}}\]hence is equal to the right-hand side
You should remember all the properties of log, as it makes sums simpler to solve. You should avoid calculation errors.