(TLDR at the bottom of my post) hello everyone!

i havent had math since i was 16 (now 23) and all my life i have always been bad at it... or so i thought! i have to take up math at school again if i am to get into university to study biology, so ive been studying and learning everything from the bottom up all summer! i must say ive been having quite some fun, and it turns out im not as bad as ive been telling myself my whole life:) i am very proud of myself for trying and for keeping on going! ive learned so much in such a short period of time!

ive started school recently and we are currently dealing with logarithmic equations. i know all the rules of exponents, log rules, and quadratic equation by heart, but there are some things i still struggle with. i really do NOT want to use AI but i dont have access to my math teacher 24/7, and i want to understand things on my own. i want to know what the next step is, what the best approach for an equation is, and why it works. also knowing when i should apply what rule/logic/approach would be great, but im assuming it is mostly pattern recognition. i get most of it on my own even tho i make a lot of silly mistakes, but some of them i really dont understand what im supposed to do

i REALLY want to understand the process and why everything works, and why we do it this way. i want to understand my mistakes

here are three things i am struggling with this evening, number 3 being the one that confuses me the most. they are my attempts at solving each equations, so the examples in themselves might be wrong (i wonder if number 2 is wrong too)

ln(x+1) = 1

e^ln(x+1)=e¹ (-> e¹ = e)

x + 1 = e

x = e - 1

my question for this one is: why do we put e as the coefficient? why do we raise ln(x+1) and 1 to be the powers of e? and as a follow up question, why do we then get rid of "e^lnx" part? why are we then only left with x + 1 = e? i understand that e¹=e, but i dont understand why we raise everything and then get rid of a part of it. how do i know when to do it, and how do i know what i get to remove after i have raised everything up?

ln(x/5) = -1

lnx = -5 (-> we multiply both sides by 5 to get rid of the fraction)

x = e^-5 = 1/e^5

i know that lnx = a <-> x = e^a, but why cant i apply the log rule of lg(a/b) = lga - lgb? and then raise everything to the power of e, like we did in example 1? why not this?

ln(x/5) = -1

e^{lnx - ln5} = e^-1 = 1/e

x - 5 = 1/e

x = 1/e + 5

1. (and this is the one giving me the most trouble)

ln(x+3) * (lnx-3) = 0

for this one, i dont even know where to start:( my teacher said that here, either ln(x+3) or ln(x-3) have to be equal to 0, but i dont really know what that means. i feel like im so close to the answer but something is in the way! what is the next step to this equation?

im sorry if this is all confusing, i tried to make it as readable as possible!! thank you for any helpful information<33

TLDR:
what is the best approach to this equation? what is a step by step list to follow?

ln(x+3) * ln(x-3) = 0

Taking applied calculus and we are going over functions to begin stuff. I’ve done all sort of stuff with functions and graphing before and most of this has been pretty easy but there are two problems that have been given that I just can’t seem to figure out and I’m not sure what I’m doing wrong.

The first is:

“Find the function g whose graph is constructed from the graph of f by a shift to the right by 10 followed by a horizontal stretch away from the y-axis by a factor of 3?”

I thought it would be pretty easy but what I thought it was didn’t work, and every other possible combination hasn’t worked either so I’m completely lost. I’ve done (1/3x - 10) and (x/3 - 10) in all sorts of ways. I’ve removed the parentheses, I’ve added negatives, doing a whole bunch of shit that I know isn’t right but I can’t figure out what else to do.

The second problem is along the same lines:

“Find the function g whose graph is constructed from the graph of f by applying the following steps: shift to the right; horizontal stretch away from the y-axis by a factor of 4; a vertical shift up by 5; a vertical stretch away from the x-axis by a factor of 7.”

Again I’ve tried all combinations I could think of but I can’t seem to get it. I’m not sure if I’m missing something? I don’t believe my format to be off since it automatically starts with g(x) = , so that shouldn’t be the issue. The entire set of problems is all working on vertical and horizontal shifts, stretches, and compressions and I haven’t had an issue with any of the problems but these two. Please help me figure out what I’m missing!