Question

How do you evaluate ${\log _3}81$?

Answer 1

Hint: It is a logarithmic equation and we have to solve it by using logarithmic theorems. These are very straightforward questions and we just have to just focus on mathematical operations while solving them.

Complete step by step solution:
According to the question we have to evaluate ${\log _3}81$, so at first, we have to understand the working of a logarithmic function. In a logarithmic function of type ${\log _b}a$, b is called as the base of log and its solution will be the power of b for which we will get $a$ as a solution. But our $a$ must be greater than zero and b must be a positive number but not equal to one, as one raised to any power is one itself. So in the question, we have the value of $a$ as 81 and the value of b as 3. So our answer will be that power of3 for which we will get 81 as a solution.
We know that ${3^2} = 9$and at the same time we also know that ${9^2} = 81$. So by these two, we can say that
$ \Rightarrow {({3^2})^2} = 81$
$ \Rightarrow {3^4} = 81$

So, we can say that our answer is 4 because three raised to the power of four is equal to 81. Hence, 4 is our answer.

Note: We have to be aware that log is basically taken as a base of 10 but any base is possible. We can also plot graphs of logarithmic functions and we will observe that its range is from negative infinity to positive infinity and the domain is all positive numbers greater than zero but not zero.