For all and for any constants such that prove that

*Proof.* We want to consider the function

If we can show this function is decreasing on the positive real axis then we establish the inequality since this would mean that if then

(So, the trick here is to think of this as a function of the exponent. The and are some positive fixed constants.) To take the derivative of we use logarithmic differentiation,

Multiplying both sides by we then obtain

Now we can conclude that for all since the first term in the product

Since (any real power of a positive number is still positive) and . For the second term we have

since and are positive, but both logarithms are negative. We know these logarithms are negative since

implies

Hence, for all . This means is a decreasing function. Therefore, if then we have