Reducing An Exponent

Question

I am reading a science book in which the author claims there are $4^{100}$ different combinations of amino acid sequences in a gene that has a length of $100$ amino acids.

No need to worry about the science here. My main concern is in the next sentence. The author says that $4^{100}$ is roughly equivalent to $10^{60}$. I am wondering, how can I verify this? How does one get from $4^{100}$ to $10^{60}$ (approximately)?

I have done lots of google searches and even asked a friend who has had plenty of high level math classes in graduate school. However, I am not really sure of what search terms to use for looking this up.

Answer 1

Saying that $4^{100} \approx 10^{60}$ is equivalent to the following statement: $100 \log 4 \approx 60 \log 10$, which you can easily verify with a calculator.

A more general fact is true. A number $a \in \mathbb N$ written in base $k \in\mathbb N$ is $(\lfloor \log_k a \rfloor + 1)$-digits long. Just substitute $a = 4^{100}$, $k = 10$ and you will get $$\lfloor \log_{10} 4^{100} \rfloor + 1 = \lfloor 100 \log_{10} 4 \rfloor + 1 = \lfloor 60.205\ldots \rfloor + 1 = 61.$$

So $4^{100}$ is $61$ digits long.

Answer 2

Hint: Compare $100\log(4)$ and $60 \log(10)$.

Answer 3

Using $2^{10} =1024 \approx 1000 = 10^3$:

$$4^{100}=2^{200}=(2^{10})^{20} \;\approx\; (10^3)^{20}=10^{60}$$

As a side note, the approximation $2^{10} \approx 10^3$ is the basis of the binary multiples where e.g. the kibi prefix stands for $2^{10}$ vs. kilo for $10^3$.