Learn How to Graph Logarithmic Equations

Prove $\frac{\ln <!--\; \; -->(x)}{x}\le <!--\; \le \; -->{\log }_{10}<!--\; \; -->2$
How would I go about proving the following inequality?
$\frac{\ln <!--\; \; -->(x)}{x}\le <!--\; \le \; -->{\log }_{10}<!--\; \; -->2$
Is there an algebraic solution and if not, why?

What are logarithms?
I have heard of logarithms, and done very little research at all. From that little bit of research I found out its in algebra 2. Sadly to say, I'm going into 9th grade, but yet I'm learning [calculus!?] and I don't know what a logarithm is! I find it in many places now. I deem it important to know what a logarithm is even though I'm jumping the gun in a sense. My understanding of concepts, is just like that of programming. In the mean time, you know its there, and your ITCHING SO HARD to find out what that is, but nope! For now we use it, tomorrow we learn what it does.
I just know that to identify a logarithm at my level, I just look for a log. :P

Solve the equations using logs:
(1.45)x =25
$2/7=(0.6{)}^{2t}$
$500=25(1.1{)}^{3x}$

Assume that a population size at time t is N(t) anf that $N(t)=40\ast <!--\; \ast \; -->2^t,t>0$
a)Find the population at time t = 0.
b)Show that $N(t)=40e^{t\ln <!--\; \; -->2},t>0$
c)How long will it take until the population size reaches 1000? [Hint:Find t so that N(t) = 1000]

How long doesit take $5,000 to grow to $10,000 at 6% compounded continuously?

Solve,
${\log }_5<!--\; \; -->(x^2+x+4)=2$

1.Find $({\log }_2<!--\; \; -->x{)}^2$ if ${\log }_2<!--\; \; -->({\log }_8<!--\; \; -->x)={\log }_8<!--\; \; -->({\log }_2)x$.
2.If $x,y>0,{\log }_y<!--\; \; -->x+{\log }_x<!--\; \; -->y=\frac{10}{3}$ and xy=144, then find $\frac{x+y}{2}$

Why does $\underset{N\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}\frac{1}{\frac{N}{1-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}}-<!--\; -\; -->i}$ converge to $\log <!--\; \; -->\left[\frac{1}{\mathrm{ϵ<!--\; ϵ\; -->}}\right]$?
while playing around with my equations, i found that the following has to hold for my universe to be consistent:
$\underset{N\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}\frac{1}{\frac{N}{1-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}}-<!--\; -\; -->i}\to <!--\; \to \; -->\log <!--\; \; -->\left[\frac{1}{\mathrm{ϵ<!--\; ϵ\; -->}}\right]\text{ for }0<\mathrm{ϵ<!--\; ϵ\; -->}<1$
playing with numerical implementations in mathematica seem to support this by "experiment", but i just don't see why.
Anybody any ideas?

Simplify: $\log <!--\; \; -->(1/1000)$ .
Write as a single logarithm and simplify (DO NOT finddecimal approximations):
$2\log <!--\; \; -->8+\log <!--\; \; -->x-<!--\; -\; -->\log <!--\; \; -->2$.
Expand and simplify (DO NOT findapproximations): $\ln <!--\; \; -->(e/x)$.
Solve for x: $\log <!--\; \; -->4+\log <!--\; \; -->(x-<!--\; -\; -->8)=2$

Suppose that In 2 = a and In 3 = b. Use properties of logarithms to write logarithm in terms of a and b.
$\ln <!--\; \; -->\sqrt[4]{\frac{2}{3}}$

Evaluate the logarithm.
${\log }_4<!--\; \; -->(16)$

If $\log$ sub(a)10 = 0.250, then $\log$ sub(10)a equals ?
Breakdown:
$\log$ sub(a)10 = 0.250
The solution requires me to the take the base(a) antilogarithmof both sides. That would be
$10=a^{0.250}$.

Find the base of x
73 (with base 8) = 214 (with base x)

I need to rewrite these statement using exponents instead oflogs and am uncertain if I did them right.
a) $\log <!--\; \; -->4=0.602$
b) $\ln <!--\; \; -->(0.646)=-<!--\; -\; -->0.437$

$3^{x+1}=27$

$\ln <!--\; \; -->(1/25)$ in terms of $\ln <!--\; \; -->5and\ln <!--\; \; -->7$

If the $\log <!--\; \; -->(x)=-<!--\; -\; -->7.65$, how do you find x?

Solve.
${\log }_a<!--\; \; -->x+{\log }_a<!--\; \; -->(x-<!--\; -\; -->2)={\log }_a<!--\; \; -->(x+4)$

Suppose that $x=\log <!--\; \; -->A$ and $y=\log <!--\; \; -->B$. i need to write these expressions in terms of x and y
$\log <!--\; \; -->(AB)$
$\log <!--\; \; -->(A^3\times <!--\; \times \; -->\sqrt{B})$
$\log <!--\; \; -->(A-<!--\; -\; -->B)$

Solve ${\log }_3<!--\; \; -->x+{\log }_4<!--\; \; -->x=4$. Please include theproperties used.