Let us look at the following logarithms:

\({\log _{10}}2 = 0.3010\), \({\log _{10}}20 = 1.3010\) and 

\({\log _{10}}200 = 2.3010\).

Characteristics of the logarithms are 0, 1 and 2. And the number of digits in the numbers 2, 20 and 200 are 1, 2 and 3 respectively. 

Thus if characteristic of logarithm of a number is \(k\), then the number will have \(k + 1\) digits. 

If \(k = 0\), then the number is a single digit number.

If \(k = - 1\), then the number is a fraction and there is no zero between the decimal point and the first non-zero digit. For example, log₁₀ (0.2) = –1 + 0.3010

If \(k = - 2\), then the number is a fraction and there is one zero between the decimal point and the first non-zero digit. For example, log₁₀ (0.02) = – 2 + 0.3010

If the logarithm of any number \(x\), (\(x\) > 1) to any positive base  \(a\), (\(a>1)\),  gives the characteristic \(n\), then we can say that the number of integers possible for \(x\) is given by \({a^{n + 1}} - {a^n}\)

Example: Find the number of digits in the number \({5^{100}}\), given that \({\log _{10}}2 = 0.3010\).

Solution: Taking log of the number we have,

\(\log \left( {{5^{100}}} \right) = 100\log 5 = 100\log \left( {\frac{{10}}{2}} \right) = 100(1 - 0.3010)\) \(= 69.90\).

Since Characteristic is \(69\), the number will have \(70\) digits.

Example 2: Find the number of digits in the expansion of \({6^{200}}\), given that \({\log _{10}}2 = 0.3010\)and \({\log _{10}}3 = 0.4771\) 

Solution: Suppose \(n = {6^{200}}\), taking logarithm in both the sides,

\({\log _{10}}n = 200{\log _{10}}6\)

\( = 200({\log _{10}}2 + {\log _{10}}3)\)

\( \Rightarrow {\log _{10}}n = 200(0.3010 + 0.4771)\)

Or \({\log _{10}}n = 155.6\)

Hence number of digits in \({6^{200}}\) = 156.

Example 3: If \({\log _{10}}x = 3.bcde....\), where \(b\) is a non-zero digit, then how many integral values are possible for \(x\)

Solution: \({\log _{10}}x = 3.\,bcde.....\). Characteristic is 3, hence the number of digits in \(x\) is 4. 

Thus \(1000 \le x < 10000\). 

The number of integral values that \(x\) can take is given by \({10^4} - {10^3} = 9000.\)