Inverses and Logs

Inverses

Its really easy to understand this subject.
Since a inverse is the reflection of a function in respect to the line y=x,
f(f^-1(x))= x.

Example: If f(x)= x^3 then to find its inverse...

f(f^-1(x))= (f^-1(x))^3. Since you would replace the input. Instead of x, you placed f^-1(x). It would be easier to write a different variable to represent f^-1(x) because it takes too long to write down. Lets have ? = f^-1(x). Since f(f^-1(x)) has to equal x in order to find its inverse, we can replace f(f^-1(x)) with x. Now your equation is x = (?)^3.

Solve for ?.

x = (?)^3                                Take the cube root of both sides.
3NTHROOT x = ?

Since we replaced f^-1(x) with ?, we can replace ? with f^-1(x) hence
3NTHROOT x = f^-1(x).


Logs

I understand this subject pretty well.

Example
Solve for x.

log₂32=x.

Step 1: If you remember the story Ms. Hwang illustrated, the log disappears and what ever number that is subscripted in the log in this case 2 is transferred over the equals sign and becomes the base of what ever number on the other side. So log₂32=x becomes 32=2^x.

From this point on there are two ways that I discovered to figure this out.

Way #1: Fast Way
Step 2: You ask yourself the question, "2 to the power of what number will give me 32?"
The answer is 5. 2 to the power of 5 will give you 32. Thus x = 5.

Way #2: Reasonable way
Step 2: You start by changing 32 into the base of 2 so that you have the same base in both sides of the equation. So you get 2^x=2^5.
Step 3: Next you remove the base 2 and make the exponents into their own bases. Since taking the log_{2 into both sides of the equation will cancel out the bases of 2 leaving the exponents as their own base. Now you get x = 5.}