Finding The Last Digit Is Easy, But What About The First Digit?

Algebra Level 4

What is the first digit of $7^{2014}$ ?

5 1 9 7 3

1 solution

Michael Mendrin
Feb 13, 2016

$2014{ Log }_{ 10 }\left( 7 \right) =1702.027453...$ , and so we have

${10}^{0.027453}...=1.065252....$ , and there's your first digit.

Can you explain this ?

Raven Herd - 5 years, 4 months ago

$\log(7^{2014}) = 1702.027 \\ \implies 7^{2014} = 10^{1702.027} = 10^{.027}\times10^{1702}=1.065252...\times10^{1072} = \boxed{1}065252...$

shaurya gupta - 5 years, 3 months ago

Wow! This means that the first few digits of 7^2014 are 1065252... That inspires another problem!

Saurabh Chaturvedi - 5 years, 4 months ago

Nice solution.

Jack Sacks - 5 years, 4 months ago