Question

The value of the expression ${{\log }_{4}}1$ is equal to:
$\begin{aligned} & \left( A \right)1 \\\ & \left( B \right)0 \\\ & \left( C \right)\infty \\\ & \left( D \right)\text{none of these} \\\ \end{aligned}$

Answer 1

The problem that we have in our hands is of logarithm. Logarithm is the inverse function of exponentiation. Although there are certain limits to logarithm, the definition is true. To solve this problem, we will assume the value of this expression as a variable and then try to solve it using other standard results. We shall proceed in this manner to get our answer.

Complete step by step solution:
We have to find the value of ${{\log }_{4}}1$. This is read as “log to the base 4, 1”. Let us first of all assign some terms that we are going to use in our solution. Let us say the value of ${{\log }_{4}}1$ is equal to ‘x’, such that now, we need to evaluate the value of this ‘x’. This can be done as follows:
We have:
$\Rightarrow {{\log }_{4}}1=x$
By the property of logarithm, we can shift the base to the right-hand side of our equation and write the term already in R.H.S. as its power. Here, the term left in the left-hand side of our equation is the number on which logarithm is being operated. On doing so, our equation becomes:
$\Rightarrow 1={{4}^{x}}$
Here, we can write the 1 in left-hand side of the equation as 4 raised to the power of zero. This means our expression becomes:
$\Rightarrow {{4}^{0}}={{4}^{x}}$
Thus, on comparison, we get the result as:
$\Rightarrow x=0$
Hence, the value of the expression ${{\log }_{4}}1$ comes out to be zero.
Hence, option (B) is the correct option.

Note: A logarithm is governed by certain rules. It can only be operated upon positive numbers. The base of a logarithm can only be positive except 1. Also, from our above solution, we can say that, “log to the base anything, 1” will always be equal to 0. This is also one of the standard properties of a logarithmic function.