Expert Answers to Algebra 2 Problems

Modeling a center of mass of a thin wire
I am asked to find the moment about the x axis for a thin wire of constant density. This thin wire lies along the curve $y=\sqrt{x}$ and the limits for integration are $x=0$ and $x=2$.
I know from my textbook that the moment about the x axis is: $M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm$
Because this is a thin wire, I know that I need to subdivide the wire into small segments for integration. I have the following for relevant data for each segment:
Length: $dl=\sqrt{x}dx$
mass: $dm=\mathrm{\delta <!--\; \delta \; -->}dl=\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}dx$
It's the part about the distance of the center of mass to the x axis that I think I'm missing. I have the following:
$\stackrel{~<!--\; ~\; -->}{y}=\sqrt{x}$
Therefore, my final integral is:
$M_y=\int <!--\; \int \; -->\stackrel{~<!--\; ~\; -->}{y}dm={\int <!--\; \int \; -->}_0^2\mathrm{\delta <!--\; \delta \; -->}\sqrt{x}\sqrt{x}dx=\mathrm{\delta <!--\; \delta \; -->}{\int <!--\; \int \; -->}_0^{2}xdx=\mathrm{\delta <!--\; \delta \; -->}{\frac{1}{2}{x}^2|}_0^2=\mathrm{\delta <!--\; \delta \; -->}2$
This particular problem is an odd numbered problem and so I know that I've got it incorrect. Please help me to see where I'm going wrong.

Equations that involve derivatives are called differential equations. Find at least one nonzero function that satisfies each of these differential equations. f'(x)=x

solve logarithmic equation without numerical methods
Is there algebraic method to solve following equation for x:
$ax+b\ln <!--\; \; -->x+c=0$
with $a,b,c$ constants without using numerical methods and ln means natural logarithm.

Lambert- W -Function calculation?
I have an equation of the form:
$n\log <!--\; \; -->n=x$
Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for it's evaluation.
Any ideas as to how I can evaluate?

Assume we have a computable, nonstandard model of Robinson arithmetic consisting of the set of integer-coefficient polynomials with positive leading coefficient, plus the zero polynomial, with their usual arithmetic.
Consider the formula
$\underset{i=0}{\overset{n}{\sum <!--\; \sum \; -->}}i=\frac{n^2+n}{2}$
Is this formula valid in this model? Let |2x| be the nonstandard natural number represented by the polynomial 2x. In this model, is the sum of all natural numbers from |0| through |2x| equal to $|2x^2+x|$? If so, does this mean some hyper-finite sums are undefined? For example, the sum of |0| through |x| would be $\frac{x^2+x}{2}$, which doesn't exist in this model.

Use of moore-penrose inverse when modeling PCA
In the derivation for principal component analysis we model our observed data points y as the result of a linear transformation restricted to being an axis change, W, applied to a set of uncorrelated variables x, where x lives in a lower dimensional space than y.
Thus, we can represent our observations as $y=Wx$, and what we are trying to find as $x=W^{\prime }y$, where W' is the pseudo-inverse of W.
Is the reason we choose to model the mapping of y onto the pca axes with use of a pseudo-inverse due to the fact that we want to be able to use PCA on systems where y doesn't completely follow our assumptions? Where not all points in y are contained in the column space of W?

Set of differential equations modeling biological system
Im trying to describe a moleculatr biological system using some differntial equations, However, differntial equations is not my strong side.
I'm thinking my equation set is quite trivial but i just cant seem to manage it. for 3 distinct groups im trying to calculate $V_1(t)$, expressed by:
$V_1(t)=V_2^{\prime }(t)k1$
$V_2(t)=V_1^{\prime }(t)k_2+V_3^{\prime }(t)k_3$
$V_3(t)=V_2^{\prime }(t)k_4$
for $k_1,k_2,k_3,k_4$ some constants.

Nonstandard models of Presburger Arithmetic
I have a question about nonstandard models of Presburger Arithmetic. I read that an example of a nonstandard model is the set of polynomials with rational coefficients with positive leading coefficient and a (positive) integer constant coefficient. It is obvious that all axioms other than the induction axiom schema are satified; but how can one prove that the induction axiom holds?

What does ${\log }_a<!--\; \; -->(b)$ equal to?
Does
${\log }_a<!--\; \; -->(b)=\frac{{\log }_c<!--\; \; -->(b)}{{\log }_c<!--\; \; -->(a)}$
or
${\log }_a<!--\; \; -->(b)=\frac{\ln <!--\; \; -->(b)}{\ln <!--\; \; -->(a)}$
??
Is there any difference between the two?

Properties of Logarithms
How do you simplify the following expression?
$\log <!--\; \; -->\left(3^{(5^7)}\right)$
I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms?
Edit: There are five answer choices:
(A) $7\log <!--\; \; -->(3^5)$
(B) $35\log <!--\; \; -->(3)$
(C) $7(\log <!--\; \; -->(3))(\log <!--\; \; -->(5))$
(D) $7(\log <!--\; \; -->(3)+\log <!--\; \; -->(5))$
(E) $5^7\log <!--\; \; -->(3)$

Basic Logarithm Equation
${\log }_2<!--\; \; -->(x)={\log }_x<!--\; \; -->(2)$
Using the change of base theorem: ${\displaystyle \frac{\log <!--\; \; -->(x)}{\log <!--\; \; -->(2)}}={\displaystyle \frac{\log <!--\; \; -->(2)}{\log <!--\; \; -->(x)}}$
Multiplied the denominators on both sides: $\log <!--\; \; -->(x)\log <!--\; \; -->(x)=\log <!--\; \; -->(2)\log <!--\; \; -->(2)$
I kind of get stuck here. I know that you can't take the square root of both sides of the equation, but still, $x=2$ seems to be an obvious solution to the equation. I've missed $2^{-<!--\; -\; -->1}$ or $\frac{1}{2}$ as another answer to the equation, which I am struggling to get to.
Any help will be greatly appreciated, thanks in advance.

Homework solving exponential equation, logarithmic equation and exponential equation.
I need help with three homework questions.¨
First one:
$\sqrt{3x^2-<!--\; -\; -->2x-<!--\; -\; -->15}=x+1$
I don't know how to get the right answer. The answer is supposed to be 4. I get:
$3x^2-<!--\; -\; -->2x-<!--\; -\; -->15=(x+1{)}^2$
$3x^2-<!--\; -\; -->2x-<!--\; -\; -->15=x^2+2x+1$
$2x^2-<!--\; -\; -->4x-<!--\; -\; -->14=0$
$x^2-<!--\; -\; -->2x-<!--\; -\; -->7=0$
$x_1=1+\sqrt{8}$
$x_2=1-<!--\; -\; -->\sqrt{8}$
Second question is:
$ln(x-<!--\; -\; -->7)-<!--\; -\; -->2ln(x-<!--\; -\; -->1)=-<!--\; -\; -->2ln5$
where i come to this stage but don't know how to solve it
$\frac{x-<!--\; -\; -->7}{(x-<!--\; -\; -->1{)}^2}=\frac{1}{25}$
the answer is supposed to be $x=11$ or $x=16$
Third question is
$9^x-<!--\; -\; -->4\ast <!--\; \ast \; -->3^x+3=0$
I have come this far
$3^{2x}-<!--\; -\; -->4\ast <!--\; \ast \; -->3^x+3=0$
answer is supposed to be $x=1$ or $x=0$

Teresa deposited S4000 into an account with 2.6% Interest, compounded semiannually. Assuming that no withdrawals are made, how much will she have in the account after 7 years?

Can $\log <!--\; \; -->(x)\log <!--\; \; -->(y)$ be reduced?
I'm currently taking Pre-Calc and am learning about logs. I know that $\log <!--\; \; -->(xy)=\log <!--\; \; -->(x)+\log <!--\; \; -->(y)$, but can $\log <!--\; \; -->(x)\log <!--\; \; -->(y)$ be reduced further?

How do you graph ${\displaystyle f\left(x\right)=\frac{2x^3+x^2-8x-4}{x^2-3x+2}}$ using holes, vertical and horizontal asymptotes, x and y intercepts?

Find number of digits of a number in another base
How can I solve this question:
Given that $\log <!--\; \; -->3$ is about $0.48$, approximately how many digits are in the number ${10}^{150}$ if it were written in base $3$
Thanks!

How to find the limit $\underset{x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}}{lim}\frac{\ln <!--\; \; -->(x^2+x+5)}{\ln <!--\; \; -->(x^8-<!--\; -\; -->x+3)}$?
I tried reducing it to the form $lim(1+x_n{)}^{1/x_n}=1$, but that didn't work.

Simple physics problem using differential equation modeling
There are 50 litres of dilution, 90% water and 10% of alcohol in a tank. Then, 50% of water and 50% of alcohol is poured in a tank, at the speed of 5 litres per minute, and the dilution is outflowing from a tank at the speed of 5 litres per minute. How much alcohol is in a tank after 10 minutes?
I have a solution that use differential equations, but I can't understand how to model the problem by differential equations. Obtained differential equation is:
$2-<!--\; -\; -->\frac{5}{50-<!--\; -\; -->t}y(t)=\frac{dy}{dt}\Rightarrow <!--\; \Rightarrow \; -->y^{\prime }+\frac{5}{50-<!--\; -\; -->t}y=-<!--\; -\; -->2$
which is a linear differential equation.
General solution:
$y(t)=C(50-<!--\; -\; -->t{)}^5+\frac{1}{2}(50-<!--\; -\; -->t)$
Now, setting $y_0=5$, which is the amount of alcohol at the beginning, we evaluate constant $C\Rightarrow <!--\; \Rightarrow \; -->$
$C=-<!--\; -\; -->\frac{20}{{50}^5}$
Going back to the general solution:
$C=-<!--\; -\; -->\frac{20}{{50}^5}$ litres.
Also, what would be the approach without using differential equations?

Simplifying trivial expression
How to simplify an expression: $e^{\frac{-<!--\; -\; -->{\ln }^2<!--\; \; -->y}{2}}$? I mean because as if $e^{\ln <!--\; \; -->y}=y$ then my guess is that this could be simplified in a similar manner thus how to do that?

Find all rational zeros of f. Then (if necessary) use the depressed equation to find all roots of the equation f(x)=0
$x^4-<!--\; -\; -->2x^3-<!--\; -\; -->43x^2-<!--\; -\; -->82x-<!--\; -\; -->24=0$