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# How many digits in the number: 2014^2022

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How many digits in the number: 2014^2022

Guest May 1, 2015

#3
+26396
+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
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#1
+26396
+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
+91411
0

Really Alan ?? :/:/

Melody  May 2, 2015
#3
+26396
+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.

.

Alan  May 2, 2015
#4
+91411
0

I am amazed.  Thanks Alan

Melody  May 2, 2015

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