Problem

Find the number of digits in the number \( 125^{100} \), without using values like \( \log_{10} 2 \) or \( \log_{10} 5 \).

Step-by-Step Derivation

1. Rewrite the Expression

\[ 125^{100} = \left(\frac{1000}{8}\right)^{100} = \frac{1000^{100}}{2^{300}} \]

Now use the fact:

\[ 2^{10} = 1024 \Rightarrow 2^{300} = 1024^{30} \]

So:

\[ 125^{100} = \frac{1000^{100}}{1024^{30}} = \frac{1000^{70}}{(1.024)^{30}} \]

2. Estimate the Denominator

We approximate:

\[ 1 < (1.024)^{30} < 10 \]

Hence:

\[ \frac{1000^{70}}{10} < 125^{100} < 1000^{70} \]

Since \( 1000^{70} = 10^{210} \), we obtain:

\[ 10^{209} < 125^{100} < 10^{210} \]

3. Count the Digits

The number of digits of a positive integer \( N \) is:

\[ \lfloor \log_{10} N \rfloor + 1 \]

Since:

\[ 125^{100} < 10^{210} \quad \text{and} \quad 125^{100} > 10^{209} \]

It follows:

\[ \log_{10} (125^{100}) \in (209, 210) \Rightarrow \lfloor \log_{10} (125^{100}) \rfloor = 209 \Rightarrow \text{Number of digits} = 209 + 1 = \boxed{210} \]

Final Answer

\[ \boxed{125^{100} \text{ has exactly } 210 \text{ digits.}} \]

Reference