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,
$\to x\log 3 = \log 1 \\\ \to x = \dfrac{{\log 1}}{{\log 3}} \;$
Now, the value of $\log 1 = 0$.Hence,
$\to x = \dfrac{0}{{\log 3}}$
$\to x = 0$
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:$\log \dfrac{a}{b} = \log a - \log b$
$\to$Power rule:$\log {a^b} = b\log a$