Find the number of digits in 2^100

Question

Find the number of digits in $2^{100}$ .

Solution — Step by Step

The number of digits in a positive integer $n$ is given by:

\text{Number of digits} = \lfloor \log_{10} n \rfloor + 1

where $\lfloor x \rfloor$ is the floor function (greatest integer $\leq x$ ).

Why? A number with $d$ digits satisfies $10^{d-1} \leq n < 10^d$ . Taking $\log_{10}$ : $d - 1 \leq \log_{10} n < d$ . So $d = \lfloor \log_{10} n \rfloor + 1$ .

\log_{10}(2^{100}) = 100 \times \log_{10} 2

Using the standard value: $\log_{10} 2 = 0.30103$

= 100 \times 0.30103 = 30.103

\lfloor 30.103 \rfloor + 1 = 30 + 1 = 31

$2^{100}$ has 31 digits.

Why This Works

The number of digits in any number $n$ counts how many times we need to multiply 10 to reach or exceed $n$ . That’s exactly what $\log_{10}(n)$ measures. The floor function takes care of the fact that we need whole digits.

For example: $\log_{10}(100) = 2$ (3 digits), $\log_{10}(999) = 2.999...$ (still 3 digits), $\log_{10}(1000) = 3$ (4 digits). The formula $\lfloor \log_{10} n \rfloor + 1$ captures this perfectly.

Alternative Method — Verification by Estimation

$2^{10} = 1024 \approx 10^3$ , so $2^{100} = (2^{10})^{10} \approx (10^3)^{10} = 10^{30}$ .

This is a number with 31 digits (the 1 followed by 30 zeros has 31 digits). Our exact calculation confirms this is 31 digits.

Common Mistake

A very common error is forgetting the $+1$ in the formula: students compute $\lfloor \log_{10}(2^{100}) \rfloor = 30$ and declare the answer to be 30 digits. The correct formula is $\lfloor \log_{10} n \rfloor + 1$ . Think about it: $10^{30}$ itself has 31 digits (a 1 followed by 30 zeros), and $\lfloor \log_{10}(10^{30}) \rfloor = 30$ , so the $+1$ is necessary to count correctly.

Question

Solution — Step by Step

Why This Works

Alternative Method — Verification by Estimation

Common Mistake

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