Indices, Exponential and Logarithmic Equations + Trig + Derivatives

[sickufully]

Yeah so this trimester I'm taking Math Methods (not by choice obviously) and since I pretty much failed what we call 'Further Maths' (the level below Methods) in yr12, I really don't see myself passing this subject unless I receive some assistance. There are some questions which have really confused me and aren't covered in the text book so if anyone knows how to solve these, please share your wisdom.

The following questions are the ones I'm having trouble with. I've attempted them but my answers just don't look right. I may post more if I get stuck again.

PS - Sorry if I didn't set my working out correctly, I'm fairly new to posting math homework online.

Indices


So for this, I wasn't sure if I first needed to switch the x^-1 and the y^-1 in the bracket and change them to positive indices OR if I took the 3 from outside the brackets and added it to each indice within the bracket. So I didn't very far with this one.


Exponential


For this, I basically just applied simple algebra rules and multiplied the first bracket by the second. So I got:
e^2x * e^2x
e^2x * -e^-x
e^-x * e^2x
e^-x * -e^-x

Collecting terms, I ended up with: e^4x -e^x

Not even sure if I did it correctly, probably wasn't even the right rule to use.

Logarithm


This isn't overly hard, I just got confused because I'm used to seeing one of the logs as just lnX. E.g. What I'm used to seeing: 3ln2 + 2ln3 - ln4.

So I've got:
= ln(2^3) + ln(3^2) - ln(4^3)
= ln8 + ln9 - ln64
= ln8 * ln9 = ln72
= ln72/ln64 = ln9/8

Am I anywhere near the answer?



I've done a question like this before but I didn't fully understand it. This is what I have so far:
= ln[(2x)^3)] - ln(4x^4) + ln[(3x^2)^2)]
= ln(8x^3) - ln(4x^4) + ln(9x^4)

I don't know what to do next, divide ln(8x^3) - ln(4x^4) OR multiply ln(4x^4) + ln(9x^4).

I think I'm supposed to divide first so I end up with ln(2x^-1) + ln(9x^4) but that doesn't seem right.

My answer ended up being: ln18x^3



This last one really threw me off as I'm not used to having the 2 in front of the log. Looking back through all lecture notes and examples, there's nothing that covers numbers in front of log.

Am I supposed to multiply something by 2 or divide by 2 or multiple the whole equation or...? WHAT DOES THIS MEAN!?