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Toilet flush at 2:07 is epic

You really overcomplicated this poor little thing! Here is how to destroy it properly. First of all, we divide everything by 5^x so we have: 1+5^x=25^x. But 25^x=(5^x)² so if we call y=5^x we have 1+y=y², which is a very known equation, since one of the solution is Phi=(1+sqrt(5))/2 called the golden number. The other solution is not interesting, since it's negative, and y=5^x>0. So we have 5^x=Phi then x.log(5)=log(Phi) then x=log(Phi)/log(5).

How can be that 49/36 is the same fraction than 42/36? There is a big error on that sequency of reasoning.

Sorry I stopped at 3:33 as I stopped understanding anything

Thanks a lot professor

Sigma

Your answer in the first exercise must be log 9 (5) + 1 = x

I have a question, this works if everything can be deconstructed to the same exponential (for example 2), but how can you solve, for example 8^x +9^x=17?

Not quite. At first you look, then you rewrite it as (2) y+y^2=y^3 with 10^x=y, so (3) y^2-Y-1=0, a tenth grade equation. Then you can ask the professor if x must be real or if you're allowed imaginary solutions--just in case (3) has negative roots. Let's assume not, take y1 the positive root ("-1" coeff indicates roots of opposite signs), and x=log(y)/log(10).