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How many digits in the number: 2014^2022

Guest May 1, 2015

Best Answer 

 #3
avatar+26971 
+5

This is best seen by examining some simple numbers.  For example:

 

There are 3 digits in 10^2  (= 100).       2*log(10) = 2  add 1 to get 3.

 

There are 4 digits in 20^3  (= 8000).      3*log(20) = 3.903...   Take integer part and add 1 to get 4

 

There are 7 digits in 35^4  (= 1500625).    4*log(35) = 6.176...   Take the integer part and add 1 to get 7

 

etc.

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Alan  May 2, 2015
 #1
avatar+26971 
+5

1). Take log to the base 10 of the number log(2014^2022) = 2022*log(2014)

$${\mathtt{2\,022}}{\mathtt{\,\times\,}}{log}_{10}\left({\mathtt{2\,014}}\right) = {\mathtt{6\,680.808\: \!240\: \!691\: \!985\: \!784\: \!6}}$$

 

2) Take the integer part (6680) and add 1 to get 6681.  That's how many digits there are.

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Alan  May 1, 2015
 #2
avatar+93305 
0

Really Alan ?? :/:/

Would you like to talk about this a bit please -it is really weird :/

Melody  May 2, 2015
 #3
avatar+26971 
+5
Best Answer

This is best seen by examining some simple numbers.  For example:

 

There are 3 digits in 10^2  (= 100).       2*log(10) = 2  add 1 to get 3.

 

There are 4 digits in 20^3  (= 8000).      3*log(20) = 3.903...   Take integer part and add 1 to get 4

 

There are 7 digits in 35^4  (= 1500625).    4*log(35) = 6.176...   Take the integer part and add 1 to get 7

 

etc.

.

Alan  May 2, 2015
 #4
avatar+93305 
0

I am amazed.  Thanks Alan  

Melody  May 2, 2015

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