Question

How do I find the logarithm \[\dfrac{{{{\log }_3}1}}{{81}}\]?

Answer 1

Hint: Here in this question, we have to find the value of the given logarithm function. First we have to find the value of \[{\log _3}1\] by using the change of base formula on getting the value of the numerator and divide that value by 81 on simplification we get the required solution.

Complete step by step answer:
The logarithm is the inverse function to exponentiation. That means the logarithm of a given number x is the exponent to which another fixed number, the base b, must be raised, to produce that number x. The logarithm function expressed as follows
\[{\log _b}x\] exactly if \[{b^y} = x\] and \[x > 0\] , \[b > 0\] and \[b \ne 1\] else if the value will be not defined.
Consider the given logarithm function:
\[ \Rightarrow \,\,\dfrac{{{{\log }_3}1}}{{81}}\]--------(1)
Here, the numerator having logarithm function i.e., \[{\log _3}1\] we can find this value by using the change-of-base formula this method can be used to evaluate a logarithm with any base.
By this, the change-of-base formula can be used to rewrite a logarithm with base n as the quotient of common or natural logs.
\[{\log _b}M = \dfrac{{\ln M}}{{\ln b}}\] and \[{\log _b}M = \dfrac{{{{\log }_n}M}}{{{{\log }_n}b}}\]
Remember the standard base value n for common \[\log \], \[\log \left( x \right)\] has base value 10 and the natural \[\log \], \[\ln (x)\] has base \[e\].
Now to evaluate the given common logarithms \[{\log _3}1\] by use the change of base formula is
For common logarithm as we know the value of new base n is 10.
\[ \Rightarrow \,\,{\log _3}1 = \dfrac{{{{\log }_{10}}1}}{{{{\log }_{10}}3}}\]
By using a logarithm calculator with base 10 the value of \[{\log _{10}}1 = 0\] and \[{\log _{10}}3 = 0.477121255\].
\[ \Rightarrow \,\,{\log _3}1 = \dfrac{0}{{0.477121255}}\]
As we know 0 divide by any number is always 0, then
\[ \Rightarrow \,\,{\log _3}1 = 0\]
On substituting the value of \[{\log _3}1\] in equation (1), then
\[ \Rightarrow \,\,\dfrac{0}{{81}}\]
\[ \Rightarrow \,\,0\]

Hence, the value of the logarithm function \[\dfrac{{{{\log }_3}1}}{{81}}\] is 0.

Note: The logarithmic function is a reciprocal or the inverse of exponential function. To solve the question, we must know about the properties of the logarithmic function. There are properties on addition, subtraction, product, division etc., on the logarithmic functions. We have to change the base of the log function and to simplify the given question.