Question

How many digits are there in the integral part of the numbers whose logarithms are respectively 4.30103, 1.4771213, 3.69897, 0.56515.

Answer 1

Hint: To solve this question, we will first of all define the logarithm and its characteristic and mantissa. And then by using the definition of characteristic, we will calculate the characteristic of all given numbers and then finally add them to get the result.

Complete step by step answer:
Let us define the logarithm and its types and characteristics and mantissa. In logarithm, the value of a positive number depends not only on the numbers but also on the base. A given positive number will have different logarithm values for different bases. There are two types of the logarithm as stated below –
(i) Natural Logarithm
(ii) Common Logarithm
The logarithm of a number to the base is known as Napierian or Natural logarithm. The logarithm of a number to the base 10 is known as a common logarithm.
The integral part of a common logarithm is called the characteristic and the non – negative decimal part is called the mantissa.
Suppose, log 39.2 = 1.5933, then 1 is the characteristic and 5933 is the mantissa of the logarithm.
Another example is, if \[\log 0.009423=-3+0.9742,\] then – 3 is the characteristic and 0.9742 is the mantissa of the logarithm.
Let us first consider, 4.30103. Then according to the above theory, we have 4 as the integral part of 4.30103.
Therefore, the number of digits = 1. Similarly, let us consider 1.4771213. Then according to the above theory, we have 1 is the integral part of 1.4771213. Therefore, the number of digits = 1.
Similarly, let us consider 3.69897. Then according to the above theory, we have 3 as the integral part of 3.69897. Therefore, the number of digits = 1.
Similarly, let us consider 3.69897. Then according to the above theory, we have 3 as the integral part of 3.69897. Therefore, the number of digits = 1.
Lastly, let us consider 0.56515. Then according to the above theory, we have 0 as the integral part of 0.56515. Therefore, the number of digits = 0.

Adding, all of them, we have the total digits as
\[\Rightarrow 1+1+1+0=3\]

Note: Let us understand the concept deeply by another example. Let us consider 0.002423, then the number can be written as 0.002423.
\[\Rightarrow 0.002423=-3+0.2423\]
As using \[0.0024=0.24\times {{10}^{-3}}\] which will give us \[{{10}^{-3}}\] as
\[\Rightarrow {{\log }_{10}}{{10}^{-3}}=-3{{\log }_{10}}10\]
\[\Rightarrow {{\log }_{10}}{{10}^{-3}}=-3\times 1\]
\[\Rightarrow {{\log }_{10}}{{10}^{-3}}=-3\]