help no clue

Can someone please help me?

THis is due tmr and I have no clue

1.

From the definition of logarithm, we obtain:

$\frac{1}{log_b{a}}=\log_a{b}$

The equation can therefore be written in the form:

$\log_n2+\log_n3+\cdots\log_n100+=\log_n(2\cdot3\cdots100)$

from which the desired conclusion immediately follows. 

2. 

https://www.desmos.com/calculator/vp8etonnna 

First, if $\sin{x}=\log{x},$  then $x\le10$ (inasmuch as $\sin{x}\le1$. 

Since $2\cdot2\pi>10$, 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 $\log{x}$ intersects the first wave of the sine curve at precisely one point. Futher, since $2\Pi+\frac{\Pi}{2}<10$, than at the point $x=\frac{5\Pi}{2}$, we have $\sin{x=1}>\log{x}$, which means that the graph of log x intersects the first half of the second positive wave of sin x. Since, at $x=10, \log{x=1}>\sin{x},$ the graph intersect this second wave another time. Therefore we conclude that the equation $\sin{x}=\log{x},$ 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

What are some of the less beautiful realms of mathematics?

1)

1/log_2(N) + 1/log_3(N) + 1/log_4(N) +..............+1/log_100(N)=1/log(100!*N)

Factor out N, then:

The above can written as:

1 / log[N * (1 x 2 x 3 x 4 x 5.................100)] = 1/ log[N*100!]