Expert Answers to Algebra 2 Problems

LN word problem
measurement of a child's ability to learn is given by the function
$$L(t)=\frac{ln(t+1)}{t+1}$$
where t is the child's age in years, for $0\le t\le 5$
At what age does a child have the greatest learning ability?
At what age is a child's learning ability increasing most rapidly?
Honestly, I'm not even sure where to start, should I take the 2nd derivative?

Logarithmic differentiation trouble with bottom of fraction
$y=\frac{(2x+3{)}^9}{\sqrt{x}(x^2-<!--\; -\; -->x{)}^6}$
I switched it to $\ln <!--\; \; -->(y)=9\ln <!--\; \; -->(2x+3)-<!--\; -\; -->6x^{1/2}\ln <!--\; \; -->(x^2-<!--\; -\; -->x)$ and then used the log rules for derivatives I know and the product rule on the right side and wound up with
$$y^{\prime }=y[\frac{18}{2x+3}-<!--\; -\; -->\frac{3\ln <!--\; \; -->(x^2-<!--\; -\; -->x)}{x^{1/2}}+\frac{6x^{1/2}(2x-<!--\; -\; -->1)}{x^2-<!--\; -\; -->x}]$$
The $\frac{18}{2x+3}$ is the correct form but the rest is way off from the form my answer is supposed to be in. I'm not quite sure what else to do with it though...

Exact SVD algorithm of matrix with symbolic entries
I am making a library with symbolic computations which supports matrices. A matrix may have symbolic entries (eg be over the ring of polynomials of a variable x ie $\mathbb{Q}[x]$).
I have implemented almost all useful linear algebra algorithms and decompositions using exact/symbolic computations (and in some cases fraction-free algorithms) (eg REF, RREF, SNF, INVERSE, PSEUDO-INVERSE, LU, QR, RANK factorisations/decompositions, ROWSPACE, COLUMNSPACE, NULLSPACE etc). The only needed algorithm which I cannot do with exact/symbolic computations is SVD/EVD. I only find numerical algorithms which solve the problem numericaly / approximately and employing "irrational" computations (eg square roots) which are not exact (note: I dont mind if an exact/symbolic algorithm employs square roots, since I can handle these symbolicaly if needed without actually computing square roots).Any exact/symbolic algorithm for SVD/EVD or any way to compute SVD using one of the decompositions I already have and which are exact?

Bob and Carol want to hire Abel Refinishing to sand and refinish the dining room floor to match the floor in the living room. Abel charges $2.79 per square foot to sand and refinish a hardwood floor. The dining room is rectangular and measures 17 feet 6 inches by 11 feet 6 inches. Find the area of the dining room floor, rounded up to the next square foot, and the cost of the work.

Complicated Logarithm
If $x>0$, $y>0$, and
$$x^2+y^2=98xy$$
then $\log <!--\; \; -->(x+y)$ can be expressed as $A\log <!--\; \; -->(x)+B\log <!--\; \; -->(y)+C$ where $A,B,C$ are real numbers and all logarithms are base $10$ logarithms. Compute $100ABC$.

Simplify: $$-<!--\; -\; -->(3m-<!--\; -\; -->6m-<!--\; -\; -->5)-<!--\; -\; -->(4-<!--\; -\; -->7m-<!--\; -\; -->2m)$$

How to solve this equation using logs
How do solve this equation for x using logarithms?
$$4^x=6^x-<!--\; -\; -->3$$
If it is not possible using logarithms, please provide another way. Thank you in advance

$\models <!--\; \models \; -->A$ vs. $A\models <!--\; \models \; -->$
As far as I understand the notion of semantic consequence (denoted by $\models <!--\; \models \; -->$), $\models <!--\; \models \; -->A$ means A is a semantic consequence of the empty set. So the "empty space" on the left side of the double turnstile means "empty set".
However, when we take a look at $A\models <!--\; \models \; -->$, now that means A is a contradiction, i.e., everything is a semantic consequence of A. Now the empty space means "everything".
Why is that? Is there any explanation for that difference?

Before leaving to visit mexico, Kevon traded 270 American Dollars and received 3,000 mexican Pesos. When he returned from Mexico, he had 100 pesos left. How much will he receive when he exchanges the pesos for American Dollars.
Kevon will receive -- dollars for his 100 pesos.

Logarithmic differentiation issue
Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on.
$$1/x(x+1)(x+2)$$
and I know that
$$log((ab)/c)=log(a)+log(b)-<!--\; -\; -->log(c)$$
So I tried to use that rule here and did:
$$ln(a)-<!--\; -\; -->ln(b)-<!--\; -\; -->ln(c)$$
and got:
$$ln(1)-<!--\; -\; -->ln(x(x+1)-<!--\; -\; -->ln(x(x+2)$$<>rbwhich simplifies to:
$$0-<!--\; -\; -->ln(x^2+x)-<!--\; -\; -->ln(x^2+2x)$$
and then I look at the solution which gives:
$$y‘<!--\; ‘\; -->=(1/(x(x+1)(x+2)))\ast <!--\; \ast \; -->(1/x+1/(x+1)+1/(x+2))$$
I'm just kind of confused on what I am doing wrong.

Value of the logarithmic expression
Why ${\log }_6<!--\; \; -->\sqrt{6}=1/2$?
I know that $\sqrt{6}=6^{1/2}$

Logarithms equation with tricky transformations
$$8^{x-<!--\; -\; -->2}\times <!--\; \times \; -->5^{x+2}=1$$
This one according to wolfram alpha it has nice solution
$$x=\frac{2(\log <!--\; \; -->(8)-<!--\; -\; -->\log <!--\; \; -->(5))}{\log <!--\; \; -->(8)+\log <!--\; \; -->(5)}$$
I see one could guess this solution and just assume left side is increasing function and be done, but I want to see some transformations which could bring me to this solution and I'm stuck.

How to clear variable $v$ from logarithmic equation
I have the following:
$6.4=-<!--\; -\; -->\log <!--\; \; -->{\displaystyle \frac{5-<!--\; -\; -->v\ast <!--\; \ast \; -->0.1}{50+v}}$
I would like to know how to solve the equation in order to get $v$'s value. Thank you very much.

A definite integral involve Logarithmic Functions
Here is the integral body:
${\int <!--\; \int \; -->}_0^m{x}^a\ln <!--\; \; -->\left(x+b\right)dx,a>-<!--\; -\; -->\frac{3}{2},b>0$

Modeling a Heat PDE
Consider a wall made of brick 10 centimeters thick, which separates a room in a house from the outside. The room is kept at 20 degrees. Initially the outside temperature is 10 degrees and the temperature in the wall has reached steady state. Then there is a sudden cold snap and the outside temperature drops to -10 degrees. Find the temperature in the wall as a function of position and time.
I am okay executing the separation of variables technique, but I can't really reason through how to model this scenario. The solution manual states that the Initial/Boundary Value Problem is...
$$u_t=k{u}_{xx},u(0,t)=20,u(10,t)=-<!--\; -\; -->10,u(x,0)=20-<!--\; -\; -->x$$
This question comes in a section before higher dimensional heat equations are introduced, but to me, it seems that this should be modeled as a three-dimensional heat equation, because thin walls are two-dimensional, and the bricks are prescribed thickness. How can I intuitively reason through this word problem to model it correctly?

Prove that a logarithm is irrational
I’m stuck with the following problem:
Prove that ${\log }_2<!--\; \; -->3\in <!--\; \in \; -->\mathbb{R}-<!--\; -\; -->\mathbb{Q}$
Thanks in advance!

I need differantial equations of this; and the type of equations $$y=c_1{e}^x+c_2{e}^{-<!--\; -\; -->x}+x$$.

Replace the Cartesian equations with equivalent polar equations. x = 7

Order of growth of logarithms, compared to linear
I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope.
Is it true that for any exponent $m>0$ (no matter how big we make $m$), the function $f(x)$
$$f(x)=(\ln <!--\; \; -->x{)}^m$$
will eventually always be less than $g(x)=x$?
I am pretty sure this is true because I have tried large values of $m$ and it always eventually slows down. But I don't know how to prove it.

Solving natural logarithms with absolute value
Question from my text: $e^{4x-<!--\; -\; -->2014}-<!--\; -\; -->7=|-<!--\; -\; -->3|$. I've never seen this before and my text is useless! Thank you!