Solveeit Logo
Login

Question

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

How do you evaluate log⁡14(14){\log _{\dfrac{1}{4}}}\left( {\dfrac{1}{4}} \right)?

Explanation

Solution

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{\log _x}(x) = 1
  • If we have a, b and c as positive integers then log⁡b(a)=c⇔bc=a{\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⁡14(14){\log _{\dfrac{1}{4}}}\left( {\dfrac{1}{4}} \right)has log base as 14\dfrac{1}{4}and log argument as 14\dfrac{1}{4}
Since both base of the logarithm and argument of the logarithm are equal i.e. are equal to 14\dfrac{1}{4}
We can say that the value of logarithm will be equal to 1

∴\therefore The value of log⁡14(14){\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⁡mn=log⁡m−log⁡n\log \dfrac{m}{n} = \log m - \log n and then they calculate values of both log base 14\dfrac{1}{4} with argument 1 and log base 14\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.