The Problem was as follows: Define log∗(n) to be the smallest number of times the log function must be iteratively applied to n to get a result less than or equal to 1. For example log∗(1000)=2 since log(1000)=3 and log(3)=0.477<=1. Let a be the smallest integer such that log∗(a)=3. Computer the number of zeroes in the base 10 representation of a

anudoneddbv

anudoneddbv

Answered question

2022-07-17

Logarithmic Contest Question
The Problem was as follows:
Define log ( n ) to be the smallest number of times the log function must be iteratively applied to n to get a result less than or equal to 1. For example log ( 1000 ) = 2 since l o g ( 1000 ) = 3 and l o g ( 3 ) = 0.477 1. Let a be the smallest integer such that log ( a ) = 3. Computer the number of zeroes in the base 10 representation of a
My answer was 10 10 but they claimed it to be 9
My logic was that log 10 10 10 = 10 10 and log 10 10 = 10 and log 10 = 1 and 1 1 so the log 10 10 10 = 3
I presume this means that a smaller answer exists, or I made a logic error somewhere. Can someone show how the smallest answer has 9 zeroes?

Answer & Explanation

autosmut6p

autosmut6p

Beginner2022-07-18Added 8 answers

10 10 10 is not the smallest correct response. Notice that 10 10 + 1 also succeeds: log log 10 10 = log 10 = 1, so log log ( 10 10 + 1 ) > 1. Hence 3 logs are needed to reduce 10000000001 to a number less than or equal to one, since two is not enough. A simple count gives 9 zeroes.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?