Das Miscellany

Musings on software, math, guitars and more ...

On a Cool Logarithm Identity

And factorials

14 May 2020

What is the value of this sum?

Well, it turns out that the answer is just . Isn’t that cool. Let’s prove it.

Prep

Let’s recall some basic logarithmic identitities. First the relationship between logs of different bases …

And then the relationship between the log of the product of two numbers and the logs of those numbers …

Proof

We are asked to show that …

We know that …

So …

And therefore …

Similarly …

So the left hand side of the original expression is …

We also know that …

So …

Plugging this back into the left hand side of the original expression we get …

Which is equal to . Therefore …

QED