Expert Answers to Algebra 2 Problems

Solve ${\log }_3<!--\; \; -->(3x+2)={\log }_9<!--\; \; -->(4x+5)$ for x
I changed the bases of the logs
$\frac{{\log }_{10}<!--\; \; -->(3x+2)}{{\log }_{10}<!--\; \; -->(3)}=\frac{{\log }_{10}<!--\; \; -->(4x+5)}{{\log }_{10}<!--\; \; -->(9)}$
Now I'm stuck, I don't know how to eliminate the logs.
On WolframAlpha I've seen that ${\displaystyle \frac{{\log }_{10}<!--\; \; -->(3x+2)}{{\log }_{10}<!--\; \; -->(3)}}=0$ gives $x=-<!--\; -\; -->\frac{1}{3}$.
Do you know how does this equation and the equation above can be solved?
EDIT: You all solved the first equation, but I still don't understand how to solve the second one (the one solved by WolframAlpha).

How do you find the zeros of ${\displaystyle f\left(x\right)=\frac{x^2+6x+9}{x^2-9}}$?

Solve the equation: $2^{2x+1}={\left(\frac{1}{32}\right)}^x$
Having trouble with this problem:
$2^{2x+1}=\frac{1}{{32}^x}$
Do I need to set the exponents equal to each other?

Multiplication using addition using logarithms
Multiplication by addition using logarithms is possible and took place in past using slide rule and log tables. Is it still used in software? Maybe sometimes it's faster to convert numbers and use addition operations instead of multiplication, then convert numbers back? Is it used somewhere? If yes, where?

Variable naming convention in mathematical modeling
Is there a guide or source of inspiration I can use when picking variable and constant names for a mathematical formula? I'm in the process of converting a poorly written technical document into something cleaner, and I'm faced with garbage like this:
$HWP\mathrm{\_<!--\; \_\; -->}cu{r}_x=Cut\mathrm{\_<!--\; \_\; -->}cur\times <!--\; \times \; -->\underset{t=0}{\overset{ru\left(\frac{yr\mathrm{\_<!--\; \_\; -->}cur-<!--\; -\; -->start\mathrm{\_<!--\; \_\; -->}date}{Freq\mathrm{\_<!--\; \_\; -->}cu{r}_x}\right)}{\sum <!--\; \sum \; -->}}f(Decay\mathrm{\_<!--\; \_\; -->}cu{r}_x;yr\mathrm{\_<!--\; \_\; -->}cur-<!--\; -\; -->start\mathrm{\_<!--\; \_\; -->}dat{e}_x)$
I have a background in scientific computing, so I can appreciate an elegantly constructed formula, but I've always wondered how a good mathematician would choose variable names.

Proof of reciprocal logarithm
I need to prove this logarithm.
$${\log }_p<!--\; \; -->(\frac{1}{x})=-<!--\; -\; -->{\log }_p<!--\; \; -->(x)$$
The first step would be $\ln <!--\; \; -->(1/x)/{\ln }_p$
I need help as to what the next step would be.

Find the value of ${\log }_8<!--\; \; -->9\times <!--\; \times \; -->{\log }_9<!--\; \; -->10\times <!--\; \times \; -->\cdots <!--\; \cdots \; -->\times <!--\; \times \; -->{\log }_n<!--\; \; -->(n+1)\times <!--\; \times \; -->{\log }_{n+1}<!--\; \; -->8$
I'm completely stumped as to where to even begin looking. If someone could point me in the right direction I'd appreciate it.

A question in interview for trinity college, Cambridge
Let $M$ be a large real number. Explain why there must be exactly one root $w$ of the equation $Mx=e^x$ with $w>1$. Why is log $M$ a reasonable approximation to $w$? Write $w=\log <!--\; \; -->M+y$. Can you give an approximation to y, and hence improve on log $M$ as an approximation to $w$?
I think for the first part perhaps a graph suffices. But I feel a bit unsure of the meaning of $\log <!--\; \; -->M$, is it $\ln <!--\; \; -->M$? or ${\log }_{10}<!--\; \; -->M$?
Thanks for your help in advance.

Solving $3^x\cdot <!--\; \cdot \; -->4^{2x+1}=6^{x+2}$
Find the exact value of x for the equation $(3^x)(4^{2x+1})=6^{x+2}$
Give your answer in the form $\frac{\ln <!--\; \; -->a}{\ln <!--\; \; -->b}$ where a and b are integers.
I have tried using a substitution method, i.e. putting $2^2$ to be $y$, but I have ended up with a complicated equation that hasn't put me any closer to the solution in correct form. Any hints or guidance are much appreciated.
Thanks

How to solve $4x-<!--\; -\; -->\log <!--\; \; -->(x)=0$
I have a problem solving this equation: $4x-<!--\; -\; -->\log <!--\; \; -->(x)=0$. I can't seem to get this equation to a simpler form featuring $\log$s only or getting rid of the $\log$
Is there a way to solve it without the Lambert-W function?

Exponential model for population
I am having trouble with a population modeling problem. The first part states:
Assume that Switzerland's population grows at a rate of 0.18 percent a year and that the 1988 population was 6.7 million.
I got the model to be:
$y(t)=(6.7\times {10}^6)e^{0.0018\ast <!--\; \ast \; -->t}$
I am having trouble with the second part of the problem:
If there is a net immigration of 10,000 people a year into Switzerland, write an expression for the population in year t.
I have tried multiple variations of adding 10000 and 10000t to different parts of the equation with no luck.
How do I account for the 10000 immigrants?

A city had a population of 4,226 at the begining of 1950 and has been growing at 8.4% per year since then. Find the size of the city at the beginning of 2004.

Solve for $x$ in: $e^{2\ln <!--\; \; -->(x)-<!--\; -\; -->\ln <!--\; \; -->(x^2+x-<!--\; -\; -->3)}=1$
I took the natural log of both sides, and simplified. Here is what I've gotten it down to:
$$2\ln <!--\; \; -->(x)=\ln <!--\; \; -->(x^2+x-<!--\; -\; -->3)$$
And I'm not sure if you can raise e to the power of each side with that 2 there. Thanks in advance.

How to solve ${\log }_2<!--\; \; -->x=x-<!--\; -\; -->1$
$$y={\log }_2<!--\; \; -->x=x-<!--\; -\; -->1$$
I thought to write
$$2^{(x-<!--\; -\; -->1)}=x$$
What should I do next?

How do I prove that $$ab+5(a-<!--\; -\; -->b)=1$$
If ${\log }_{12}<!--\; \; -->18=a$ and ${\log }_{24}<!--\; \; -->54=b$ then how do I prove $ab+5(a-<!--\; -\; -->b)=1$?
I figured that out it's ${\log }_a<!--\; \; -->b$ and ${\log }_{2a}<!--\; \; -->3b$ but how do I solve it?

Equation with logarithms and absolute values
I have this equation and I want to solve it for $x<0$
$$\frac{\ln <!--\; \; -->|x|}{|x|}=\frac{\ln <!--\; \; -->|x|}{x}$$
According to WolframAlpha, the solution is $x=-<!--\; -\; -->1$ but I don't know how to get that.
My approach:
$$\frac{\ln <!--\; \; -->|x|}{|x|}=\frac{\ln <!--\; \; -->|x|}{x}$$
$$\frac{1}{|x|}=\frac{1}{x}$$
$$|x|=x$$
$$-<!--\; -\; -->x=x$$
$$2x=0$$
$$x=0$$
which is apparently wrong. Where have I made the mistake?

Reading the mind of Prof. John Coates (motive behind his statement)
To start with the issue, I have been thinking from many days that Birch-Swinnerton-dyer conjectures should have some association with the Galois theory, but one day I got the Article of Tate called as
" J. Tate, On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue, Sem. Bonrbaki 18 (1966)",
(please don't ask me to mention the reference of the above article, I have missed the link, I have only soft-copy with me) after a deep internet search, but the article was very technical and very sophisticated, rather after going through it dozens of times, I understood that Prof. J. Tate was trying to convey that one must relate the L-function to some Galois-groups, that was the rough picture in my mind! .
But later luckily to my surprise it was a coincidence that in an work of John Coates(John Coates, The Arithmetic of Elliptic Curves with Complex Multiplication, Proceedings of the International Congress of Mathematicians Helsinki, 1978), I encountered the same words where Coates describe that
".....They also gave a conjectural formulathe coefficient of $(s-<!--\; -\; -->1{)}^{g_Q}$ ($g_Q$ is rank) in the expansion of L(E,s) about $s=1$ but we shall not discuss this here.
Now Tate's work on the geometric analogue suggests that, in order to attack this conjecture, one must relate L(E,s) to the characteristic polynomial of some canonical element in a representation of a certain Galois group.
Now I request anyone who is currently working in that area/ know that area, answer me what does the above statement mean.
i.e. what was the intuition behind linking the L(E,s) to the characteristic polynomial of some canonical element in a representation of a certain Galois group ? .
And why does one need to look at characteristic polynomial of some element in Galois group in order to proceed with the Birch and Swinnerton Dyer conjecture ? .
Can anyone give a detailed summary of what was the Tate's idea and in which sense Coates refer to that sentence ? .
Please frame your answer not in a high-technical manner, but in the way a beginner can understand, but please answer me in a detailed manner.
I hope this question doesn't go unanswered, and I get the answer in a detailed form(describing to the maximum extent). Please help me.

How to solve common log problem given the following with the help of calculator?
the question is
$${\log }_{10}<!--\; \; -->(x+8)+6=8$$
$${\log }_{10}<!--\; \; -->(x+8)=2$$
$${10}^2=x+8$$
I just turned it into exponential form. Now my question is if i did this right at all? and if i did how would i go on solving it?

Logarithm with nth root
I made it but the result is very strange. I want every step to the result
$$6{\log }_{10}<!--\; \; -->\frac{\sqrt{2}}{\sqrt[3]{3+\sqrt{5}}}$$

What are the products of real solutions of this equation?
How can I solve ${\log }_{1/2}^2<!--\; \; -->(4x)+{\log }_2\left(\frac{x^2}{8}\right)=8$?
I have tried the elementary for logarithms simplifying the terms in brackets.