1.
From the definition of logarithm, we obtain:
1logba=logab
The equation can therefore be written in the form:
logn2+logn3+⋯logn100+=logn(2⋅3⋯100)
from which the desired conclusion immediately follows.
2.
https://www.desmos.com/calculator/vp8etonnna
First, if sinx=logx, then x≤10 (inasmuch as sinx≤1.
Since 2⋅2π>10, the interval on the x-axis between x = 0 and x = 10 contains one complete period of the sine curve plug part of a second period. The graph of logx intersects the first wave of the sine curve at precisely one point. Futher, since 2Π+Π2<10, than at the point x=5Π2, we have sinx=1>logx, which means that the graph of log x intersects the first half of the second positive wave of sin x. Since, at x=10,logx=1>sinx, the graph intersect this second wave another time. Therefore we conclude that the equation sinx=logx, has exactly three roots.
Wow! I haven't done these in SO long! Thanks for bringing me back to the more beautiful realm of mathematics.
I hope this helped,
Gavin