Question

How do you evaluate \[{\log _{\dfrac{1}{4}}}\left( {\dfrac{1}{4}} \right)\]?

Answer 1

Hint: We use the property of logarithm that when the base of the log is equal to the arguments of the logarithm then the value of the logarithm equals to 1.
* Identity rule of logarithm states that if base is same as the argument then: \[{\log _x}(x) = 1\]
* If we have a, b and c as positive integers then \[{\log _b}(a) = c \Leftrightarrow {b^c} = a\]
* Base of log value is the number being raised to a power. It is simply the value that is written along with log in the subscript.
* Argument of log value is that number that is written inside the bracket.

Complete step-by-step answer:
The value in the subscript is called the base of the logarithm whereas the value inside the parentheses is called the argument of the logarithm.
Here \[{\log _{\dfrac{1}{4}}}\left( {\dfrac{1}{4}} \right)\]has log base as \[\dfrac{1}{4}\]and log argument as \[\dfrac{1}{4}\]
Since both base of the logarithm and argument of the logarithm are equal i.e. are equal to \[\dfrac{1}{4}\]
We can say that the value of logarithm will be equal to 1

\[\therefore \]The value of \[{\log _{\dfrac{1}{4}}}\left( {\dfrac{1}{4}} \right)\] will be equal to 1.

Note:
Many students make mistake of opening the logarithm here using the property \[\log \dfrac{m}{n} = \log m - \log n\] and then they calculate values of both log base \[\dfrac{1}{4}\] with argument 1 and log base \[\dfrac{1}{4}\]with argument 4 using the tables. Keep in mind we don’t need to solve such log values where the base is the same as the argument, we can directly apply the property, if the base was not specified here then we could’ve gone for the property that breaks division of log to subtraction.