Find number of digits of a number in another base How can I solve this question: Given that log3 is about 0.48, approximately how many digits are in the number 10^(150) if it were written in base 3.

vedentst9i

vedentst9i

Answered question

2022-11-14

Find number of digits of a number in another base
How can I solve this question:
Given that log 3 is about 0.48, approximately how many digits are in the number 10 150 if it were written in base 3
Thanks!

Answer & Explanation

Julius Haley

Julius Haley

Beginner2022-11-15Added 19 answers

Here is a simple solution.
Lets assume that the number of digits in the base 3 representation of 10 150 is k
So, 10 150 = 3 k 1 a k 1 + 3 k 2 a k 2 + + 3 a 1 + a 0 where 0 a i 2 where a k 1 is not 0
Now lets try to put some bounds on the R.H.S. of above.
From the above equation we have 10 150 3 k 1
Also, 3 k 1 a k 1 + 3 k 2 a k 2 + + 3 a 1 + a 0 2 ( 3 k 1 + 3 k 2 + + 3 + 1 ) = 2 ( 3 k 1 3 1 ) < 3 k
So, 3 k > 10 150 3 k 1
Since log 3 ( x ) is an increasing function we have,
k > 150 log 3 ( 10 ) k 1
So, k = 1 + 150 log 3 ( 10 )
The above method of proof clearly points at a general result. What is that?

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