Inverse Functions--Trigonometric and Logarithms--HELP?

Question

*Can you please explain some of the relationships between log and natural log...I'm confused about when I use natural log in expressions like the inverse of f(x)=e^x/7e^x?
*Also, how do I find the exact value of each expression such as arctan(sqrt3), or sin-1(-sqrt3/2)?
Explanations on this concept in general are welcome, too!

Can't thank you enough.

Answer 1

use natural log if the expression contains "e"
"
example y =(e^x-1)/e^x
Put e^x = z so y =((z-1)/z
zy= =z-1 so z = 1/(1-y) and x= lnz= -ln(1-y)
interchange x and y so y= -ln(1-x) is the inverse function
arctan(sqrt3) .
arc tan x is a function with range in the interval -pi/2 the angle whose tangent is sqrt3 is this interval is pi/3
arc sin x is a function with range -pi/2<=x<=pi/2
-sqrt3/2
belongs to an angle = -pi/3
For completing the answer
arc cos x has range 0<=x<=pi
so arc cos -sqrt2)/2 = 3pi/4

Answer 2

the natural log is a logarithm with base e. this logarithm comes up in so many situations that is was given the special name of natural log. Anytime you need to use a logarithm you should pick the log with the base that will simplify the expression the best.


for the inverse trig functions, look at the unit circle to help find the answers.

arctan(Sqrt(3)) = x such that tan(x) = Sqrt(3)
x = 60° or 240°

arcsin(-Sqrt(3)/2) = x such that sin(x) = -Sqrt(3)/2
x = 210°, 300°, or -60°

look at the unit circle and also be aware of the range for which the answer is needed in. there are conventions, but be careful on your ansignments.