Question

Find the value of ${{\log }_{1}}1$.

Answer 1

- Hint:First we should understand the term ‘logarithm’ and then we will see the property of logarithm required to solve this question. Remember the domains in which the value of a logarithmic function is defined.

Complete step-by-step solution -
In mathematics, the logarithm is the inverse function of exponentiation. That means that the logarithm of a given number ‘n’ is the exponent to which another fixed number the base ‘b’ must be raised, to produce that number ‘n’. Common logarithm has base 10, however we can convert it to any number. Let us take an example: consider a number, here I am using 100, so, 100 can be written as 10 raised to the power 2 or mathematically, ${{10}^{2}}$. Now, we have to find the logarithmic value of 100 with 10 as considering the base of the logarithm. In other words, we can interpret the question as ‘to how much must be the power of 10 should be raised, so that it becomes equal to 100’. We know that 10 raised to power 2 is equal to 100, so the answer is 2. Mathematically, it can be written as ${{\log }_{10}}100=2$. Some important formulas for logarithms are:
$\begin{aligned} & {{\log }_{m}}{{n}^{a}}=a{{\log }_{m}}n,\text{ } \\\ & {{\log }_{a}}\left( m\times n \right)={{\log }_{a}}m+{{\log }_{a}}n\text{ } \\\ & \text{lo}{{\text{g}}_{a}}\left( \dfrac{m}{n} \right)={{\log }_{a}}m-{{\log }_{a}}n \\\ & {{\log }_{{{a}^{b}}}}m=\dfrac{1}{b}{{\log }_{a}}m \\\ \end{aligned}$
${{\log }_{n}}m=\dfrac{\ln m}{\ln n}$
It is important to note that logarithmic function is not defined for negative values of base and base 1. ${{1}^{n}}=1$ has infinite solutions from which we can pick any value to be its solution. Thus, we can see that logarithm is not well defined for base 1.
${{\log }_{1}}1$ can be written as $\dfrac{\ln 1}{\ln 1}$, using change of Base rule, which is of the form $\dfrac{0}{0}$, and hence it is indeterminate. In this way also we can prove that ${{\log }_{1}}1$ is an undefined function.

Note: One may get confused that the value of log 1 is 0, but you have to remember the domain in which the logarithmic function is defined. Logarithmic function is defined for all the values of base except 1 and negative values, so for ${{\log }_{a}}b$ to be defined ‘a’ must be greater than 0 and unequal to 1 and also, b must be greater than 0. Hence, the value of ${{\log }_{1}}1$ cannot be 0 and is undefined.