Question

Find the value of log 1 to the base 3?

Answer 1

Hint: We know that the value for log 1 to the base 10 is 0, but le us check what will be its value for base 3. For that, let ${\log }_{3}1$ be equal to x. Now, ${\log }_{3}1=x$ can be written as $3^x=1$. Now, introduce log on both sides and use the property $\log a^x=x\log a$ and simplify, we will get the value of x.

Complete step-by-step answer:
In this question, we have to find the value of log 1 to the base 3.
Now, we know that the value of log 1 to the base 10 is 0, but let us check what will be its value for base 3.
Let the log of 1 with base 3 be x. Hence, we can write it in equation form as
$\to {\log }_{3}1=x$- - - - - - - - - - - (1)
Now, this is of the form ${\log }_{a}b=x$, and we know the property of log that when a log of a number is given with base, we can write it as base raise to the value of log equal to the number. Hence,
 If ${\log }_{a}b=x$ , then $a^x=b$.
Using this property in equation (1), we get
$\to 3^x=1$
Now, introduce log on both sides, ass we need to find the value of x
$\to \log 3^x=\log 1$
Now, we have another property of log that is $\log a^x=x\log a$. Therefore,
$$\begin{gathered} \to x\log 3=\log 1 \\ \to x={\displaystyle \frac{\log 1}{\log 3}} \end{gathered}$$
Now, the value of $\log 1=0$.Hence,
$\to x={\displaystyle \frac{0}{\log 3}}$
$\to x=0$
Hence, the value of log 1 to the base 3 is 0.
So, the correct answer is “0”.

Note: The value of log 1 to any base will always be equal to 0 only. Also, note that the value of log 0 to any base will always be equal to 1. Some of the important properties of logs are:
$\to$Product rule: $\log ab=\log a+\log b$
$\to$Division rule:$\log {\displaystyle \frac{a}{b}}=\log a-\log b$
$\to$Power rule:$\log a^b=b\log a$