Question

How do you evaluate \[\dfrac{{{{\log }_4}1}}{{64}}\]?

Answer 1

Hint: In the given question, we have been given an expression. This expression contains a function, which is the logarithm function. We have to solve the logarithm when we have been given the base and the argument of the logarithm. We do that by expressing the argument in terms of the given base and solving it by using the appropriate formula.

Formula Used:
We are going to use the formula of logarithm, which is:
\[{\log _b}a = n \Rightarrow {b^n} = a\]

Complete step by step answer:
The given expression is \[\dfrac{{{{\log }_4}1}}{{64}}\]. Let \[p = {\log _4}1\]. First, we have to solve for the value of \[p\].
The basic formula of logarithm says,
If \[{\log _b}a = n\]
then, \[{b^n} = a\]
Hence, putting \[b = 4\], \[a = 1\], we get,
\[{4^p} = 1\]
We know, \[{4^0} = 1\].
Hence, \[{4^p} = {4^0}\]
Since the bases are equal, we can equate the powers,
\[p = 0\]
Hence, \[\dfrac{{{{\log }_4}1}}{{64}} = \dfrac{p}{{64}} = \dfrac{0}{{64}} = 0\].

Thus, the answer to the question \[\dfrac{{{{\log }_4}1}}{{64}}\] is \[0\].

Additional Information:
The \[\log \] function has other basic properties too:
\[{\log _x}{x^n} = n\]
\[{\log _a}b = \dfrac{1}{{{{\log }_b}a}}\]

Note:
In the given question, we had been given a logarithmic expression. We had to solve the logarithm when we were given the base and the argument of the logarithm. We do that by expressing the argument in terms of the given base and solving it by using the appropriate formula. So, it is really important that we know the formulae and where, when, and how to use them so that we can get the correct result.