How can i count number of digits in tetrated numbers?

Question

As you read in the title, I need a technique for counting number of digits in tetrated numbers.

For example:

${3^{(3)}}^3 = 7625597484987 $(13 digits)

${7^{(7)}}^7 = $How many digits (approximately)?

${{11}^{(11)}}^{11} $= How many digits (approximately)?

Is there any technique for calculation?

Answer 1

The number of digits of $n$ is $\lfloor \log_{10}n\rfloor+1$ If we ignore the floor and the $+1$, we get $\log_{10}7^{7^7}=7^7 \log 7\approx 695974$ as shown by Alpha. This is fine for three layer tetrations, but is inadequate for taller ones. For example, if you ask Alpha for $7^{7^7}\log_{10}7$ you get back $10^{10^{5.842593328962333}}$ which is still in tetrated form. Basically each log only takes one layer off the stack.