Prove log(x)^(10) < x (for x>10^(10))

tikaj1x

tikaj1x

Answered question

2022-10-15

Prove log ( x ) 10 < x (for x > 10 10 )
I need to prove that log ( x ) 10 < x for   x > 10 10 It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.

Answer & Explanation

Crubery3

Crubery3

Beginner2022-10-16Added 6 answers

Assuming this is log base 10, you know that log 10 ( 10 n ) 10 = n 10 (this follows from the identity log a ( a k ) = k), so it seems you would have to verify that 10 n > n 10 for n > 10. I am not sure that this makes it easier for you, but atleast the log is gone. You can then show that the derivative of n 10 grows slower than 10 n
faois3nh

faois3nh

Beginner2022-10-17Added 4 answers

You want, for x > 10 10 , ( log ( x ) ) 10 < x. Setting x = y 10 , this is ( log ( y 10 ) ) 10 < y 10 for y > 10 or log ( y 10 ) < y or y 10 < 10 y .
Since x 1 / x is decreasing for x > e, if y > 10, 10 1 / 10 > y 1 / y or 10 y > y 10

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