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24^25 or 25^24: Proving Which Is Larger

Date: 03/12/2015 at 07:26:27
From: Gerry
Subject: Inequality

How would you prove that 24^25 > 25^24 without using a calculator?

I thought of mathematical induction, but it's not the case that
(a + 1)^a < a^(a + 1).

Thank you.

Date: 03/12/2015 at 08:37:02
From: Doctor Ali
Subject: Re: Inequality

Hi Gerry!

Thanks for writing to Dr. Math.


     A1 = 24^25
     B1 = 25^24

To begin, let's raise both numbers to the power 1/(24*25). This gives

     A2 = 24^(1/24)
     B2 = 25^(1/25)

Now, if we show that A2 is greater than B2, we can easily deduce that A1 
is also greater than B1.

To do this, we can consider

     y = x^(1/x)

Taking the derivative of both sides gives

           (1 - ln(x)) x^(1/x)
     y' = ---------------------

We can clearly see that y' is always a negative number when x is greater 
than e, which is about 2.718281. This means that this function is 
decreasing for x > e. And this in turn means that

     B2 < A2


     B1 < A1


     24^25 > 25^24

Does that make sense?

For some similar problems, please take a look at the following pages from
our archive: 

I hope it helps.

Please write back if you still have any difficulties.

- Doctor Ali, The Math Forum 

Date: 03/12/2015 at 09:00:37
From: Gerry
Subject: Thank you (Inequality)

Thank you!
Associated Topics:
High School Calculus
High School Exponents
High School Number Theory

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