In Mathematica Log[z] gives the natural logarithm of z (logarithm to base e) and Log[b,z] gives the logarithm to base b.

Log[512]//PowerExpand
(*  9 Log[2] *)

Dear Mariusz Iwaniuk,

Thank you so much for your response. It was a great help :) However, for the second part where you mentioned removing the comma, I get a different answer when I do the same thing by hand. When I do it manually, I get the answer as "9 ln 2" instead of "Log [512]"

I realise that 9 ln 2 = ln 512, but aren't Log and Ln different?

Sorry for the confusion, I'm a complete beginner to Mathematica.

Thank you so much, Raivat S.

Hello

First integral can't be solved symbolically. Integral with gamma function has no antiderivative with standard elementary function or maybe known special function, no discovery yet. Use NIntegrate,a numericall integrator.

 NIntegrate[Gamma[n + 1]/Gamma[k + 1]*1/Gamma[n - k + 1]*(1/10000)^k, {n, 25,  500}, {k, 2, 25}]
 (* 0.0481051 *)

Second integral, you did little mistake in syntax.

Integrate[(x*y^2)*(1/(x^2 + 1)), {x, 0, 1}, {y, -3, 3}]
(* Log[512] *)

Regards Mariusz

PS: Next time post code not image.SEE