Difficult Maths/Physics Problems Help Thread

Made this one up last academic year:

Find $\dfrac{dy}{dx}$ if $y=\dfrac{\log_{50}{x}}{\log_{50}{20}}$

That's correct, the base could be Graham's number and you'd still get the same result

Ahaha yes This may be a shot in the dark but fellow numberphile subscriber here?

Made this one up last academic year:

Find $\dfrac{dy}{dx}$ if $y=\dfrac{\log_{50}{x}}{\log_{50}{20}}$

What's bigger, Graham's number to the googolplex, or googolplex to the Graham's number?

Edit: Whoops, didn't realise everyone else had done it!

I'm going to guess googolplex to Graham's number because if we compare x^y and y^x, log both sides we get y*lnx and x*lny. If y>x and x,y>1 (maybe also x,y>e), it looks like x^y is at least as large.

That could be complete rubbish though.

Also that number would be massive

I was quite happy when I solved this, because I usually find these questions very hard.


If $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfies

1)$f(1)=2$
2)$\displaystyle \forall x,y\in\mathbb{Q}, f(xy)=f(x)f(y)-f(x+y)+1$

Find $f(x)$

I am having some difficulty finding a combinatorial proof of the identity

(n+1)C(k+1)=((n+1)/(k+1))*nCk where C is the choose symbol.

any suggestions ?

I was quite happy when I solved this, because I usually find these questions very hard.


If $f: \mathbb{Q} \rightarrow \mathbb{Q}$ satisfies

1)$f(1)=2$
2)$\displaystyle \forall x,y\in\mathbb{Q}, f(xy)=f(x)f(y)-f(x+y)+1$

Find $f(x)$

What do you mean by combinatorial?
Do you mean by relating it to a 'real' problem?

If you do it via factorials it comes out straight away.

Sketch the graph of $y=\log_1x$ (not that hard but fairly interesting)



That's a good way of looking at it, because as $x\rightarrow\infty$, y increases much more slowly for $\ln{x}$ than a scalar increase, if that makes sense.

If you put ln(100000000000000000000000000) in your calculator it is surprisingly small.