Number sequence with a given formula

Question

$a_{n}$ is a sequence which is given by the formula: $a_{n+1}=\log_2(a_n)$, where $a_1=30$. What is a number of maximum elements of the sequence? I tried to approximate each number but I think there must be more systematic way to solve this problem. Can you explain this to me? Note that sequence only contains real numbers.

Answer 1

$$16<30<65536.$$

Taking the logarithm once,

$$4<\log_230<16.$$

Twice,

$$2<\log_2\log_230<4.$$

Thrice,

$$1<\log_2\log_2\log_230<2.$$

Four times,

$$0<\log_2\log_2\log_2\log_230<1.$$

Five times,

$$\log_2\log_2\log_2\log_2\log_230<0.$$

Answer 2

Note sure what you are asking. If you are assuming $a_n \in \mathbb{C}$, then the sequence has an infinite number of terms (but you have to define the log branch you are using).

If you are assuming $a_n \in \mathbb{R}$, note the terms are decreasing quickly. What happens to $a_{n+1}$ when $a_n < 1$?

Answer 3

Notice that

$$a_2 = \log_2(a_1)$$ $$\implies 4<a_2<5$$ Taking the logarithm of each term, $$\implies \log_2 4 < \log_2(a_2)<\log_25$$ $$\implies2<a_3<3$$ (As $\log_25 <3$)

If you keep taking the logarithm, you'll finally get: $$-\infty <a_6<0$$

So $a_6$ will be the last term in the series.