Question

How do solve $\log 1$ ?

Answer 1

Hint : A logarithm is the power that you raise a certain base to, in order to get a given number. Logarithms are like a way to inverse exponents, here to solve the question by different methods, basically the first method by using a standard logarithm table and also solved by standard logarithmic identities.

Complete step-by-step answer :
In mathematics, 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
${\log _b}\left( x \right) = y$ exactly if ${b^y} = x$ and $x > 0$ and $b > 0$ and $b \ne 1$ else if $b = 1$ the value will be not defined
The log function with base $b = 10$ is called the common logarithmic function and it is denoted by ${\log _{10}}$ or simply written as $\log$ .
The log function having base $e$ is called the natural logarithmic function and it is denoted by ${\log _e}$ .
These two bases are used commonly in logarithm functions.
Method 1
To find the logarithm of a number, basically we can use the logarithm table instead of using a mere calculation.
While using the logarithm table first we understand the characteristic and mantissa part.
Characteristic part - The whole part of a number is called the characteristic part. The characteristic of any number greater than one is positive, and if it is one less than the number of digits to the left of the decimal point in a given number. If the number is less than one, the characteristic is negative and is one more than the number of zeros to the right of the decimal point.
Mantissa part: The decimal part of the logarithm number is said to be the mantissa part and it should always be a positive value. If the mantissa part is in a negative value, then convert into the positive value.
Procedure to solve the given question $\log 1$ using logarithm table:
I.Every logarithm table is only usable with a certain base. The given logarithm function has base 10
II.The characteristic part of $\log 1$ is 1 and mantissa part is 0
III.The logarithm stable start with characteristic part 10 below that values will be zero or the characteristic part of log up to 10 is zero
Hence the value of $\log 1 = 0$ .

Method 2
As we know some standard identities of logarithms I.e.,
1. $\log {m^n} = n\log m$
2. $\log \left( {\dfrac{m}{n}} \right) = \log m - \log n$
Consider $\log 1$ can be written as
$\Rightarrow \,\,\log {1^0}$ where, ${1^0} = 1$
$\therefore \,\,\log {1^0} = 0\log 1$ [Anything $\times$ 0 = 0]
$\therefore \,\,\log 1 = 0$
Also $\log 1$ can be written as
$\Rightarrow \,\,\log \left( {\dfrac{1}{1}} \right)$
$\therefore \,\,\,\,\log \left( {\dfrac{1}{1}} \right) = \log 1 - \log 1$ $\left[ {\because \,\,\,\log a - \log a = 0} \right]$
$\therefore \,\,\log 1 = 0$
Hence by all the methods of solving the value of $\,\log 1 = 0$
So, the correct answer is “0”.

Note : If the function contains the log term, then the function is known as logarithmic function. We have two types of logarithms namely common logarithm and natural logarithm. The two logarithms have different values for the numerals. So to solve the logarithmic function we need to know about the base.