27.15. How many digits does the number $2^{100}$ have?

## 27.3. Identities for logarithms
27.15. Answer: 31 digits. It is clear that $2^{100}=1024^{10}>1000^{10}$, so the number $2^{100}$ has no fewer than 31 digits. On the other hand,

$$
\frac{1024^{10}}{1000^{10}}<\left(\frac{1025}{1000}\right)^{10}=\left(\frac{41}{40}\right)^{10}<\frac{41}{40} \cdot \frac{40}{39} \cdot \frac{39}{38} \cdot \ldots \cdot \frac{32}{31}=\frac{41}{31}<10
$$

Thus, $2^{100}=1024^{10}<10 \cdot 1000^{10}$, so the number $2^{100}$ has fewer than 32 digits.