Algebra Word Problem Solutions

Find two consecutive odd integers whose sum is 36

Write an equation in point-slope form for the given (-5, 0) with slope 4

Why does the branch cut for log(1+z), cutting away the negative axis, start at -1?
I am really confused about this concept.
Is it because from -1 to 0, the inputs (1+z) are just viewed as inputs that aren't actually from the negative axis at all, and so log(1+z) is well-defined and continuous on (-1,0]? I don't see the single-valuedness on (-1,0], though. Shouldn't log(1+z) jump by a $2\mathrm{\pi <!--\; \pi \; -->}i$ term, on (-1,0]?
Any comments are welcome.
Thanks,

${\displaystyle x^4-12x^2+11<0}$

Write an equation in slope-intercept form for a line parallel to y = -5 that passes through the point (-1,-8)

Are there any irrational numbers that have a difference of a rational number?

How to prove "Logarithms grow more slowly than polynomials"
"Logarithms grow more slowly than polynomials. That is, $\Theta (\lg <!--\; \; -->n)$ grows more slowly than $\frac{-<!--\; -\; -->b\pm <!--\; \pm \; -->\sqrt{b^2-<!--\; -\; -->4ac}}{2a}$ for any positive constant a."
if $y_1=x^{1/2}$) and $y_2=\log <!--\; \; -->(x)$
by comparing the graphs can say $y_1>y_2$
But, is there a way to prove this?

logarithmic Series
I'm aware that by properties of logarithm
$\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}\ln <!--\; \; -->(k)=\ln <!--\; \; -->(n!)$
My question is if
$\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}{\ln }^2<!--\; \; -->(k)={\ln }^2<!--\; \; -->(n!)?$
Because when I am verifying the value where $n=5$, I get different result... maybe Im missing something... is there a formula that defines
$\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}{\ln }^2<!--\; \; -->(k)\text{ ?}$
Here's my computation:
$\underset{k=1}{\overset{n}{\sum <!--\; \sum \; -->}}{\ln }^2<!--\; \; -->(k)={\ln }^2<!--\; \; -->(n!)$
$\underset{k=1}{\overset{5}{\sum <!--\; \sum \; -->}}{\ln }^2<!--\; \; -->(k)={\ln }^2<!--\; \; -->(5!)$
${\ln }^2<!--\; \; -->(1)+{\ln }^2<!--\; \; -->(2)+{\ln }^2<!--\; \; -->(3)+{\ln }^2<!--\; \; -->(4)+{\ln }^2<!--\; \; -->(5)={\ln }^2<!--\; \; -->(120)$
$0+0.48045+1.20694+1.92181+2.59029=22.92007$
$6.199494$ is not equal to $22.92007$

Write an equation of a line passing through (6, 3), perpendicular to y=2x+2

Write an equation in point-slope form for the given (-3,-5) Slope: -5

Write the first five terms of the geometric sequence ${\displaystyle a_1=1,r=\frac{1}{3}}$

Find the slope of a line parallel to 3x`+y=15

I need help solving a logarithm equation.
these were my steps. Can someone tell me where have i goe wrong since the answer is $-<!--\; -\; -->\frac{1}{4}$
$\sqrt{\log <!--\; \; -->(\sqrt{10}a)}=\frac{1}{2}$
$\frac{1}{2}\log <!--\; \; -->(\sqrt{10}a)=\log <!--\; \; -->\sqrt{10}$
${\log <!--\; \; -->(\sqrt{10}a)}^{\frac{1}{2}}=\log <!--\; \; -->\sqrt{1}0$ because $({\log }_a<!--\; \; -->x^n=n{\log }_a<!--\; \; -->x)$
$(\sqrt{10}a{)}^{\frac{1}{2}}={10}^{\frac{1}{2}}$ (squaring)
${10}^{\frac{1}{2}}a=10$ division by ${10}^{\frac{1}{2}}$
$a={10}^{1-<!--\; -\; -->\frac{1}{2}}$
$a=\sqrt{10}$

Let $\mathrm{\tau <!--\; \tau \; -->}$ be a topology on a set $X$ and consider $A\subseteq <!--\; \subseteq \; -->X$. If $x\in <!--\; \in \; -->X$ is an accumulation point of $A$ if:
$\mathrm{\forall <!--\; \forall \; -->}\mathcal{U}\in <!--\; \in \; -->\mathrm{\tau <!--\; \tau \; -->},x\in <!--\; \in \; -->\mathcal{U}⟹<!--\; ⟹\; -->A\cap <!--\; \cap \; -->\mathcal{U}$ contains another point other than $x$.

How can i know if two lines are parallel?

Limit evaluate $\underset{x\to <!--\; \to \; -->0}{lim}\frac{\ln <!--\; \; -->(\cos <!--\; \; -->(-<!--\; -\; -->5x))}{\ln <!--\; \; -->(\cos <!--\; \; -->(-<!--\; -\; -->3x))}$
Please help me with this limit without using L'Hôpital's rule. I would by happy if you use simple solving. Thank you as much as I can ;).

Write an equation in point slope form that passes through P(-2,4) and parallel to x=5

Infinite inequality with logarithm
Solve inequality $3-<!--\; -\; -->{\log }_{0.5}<!--\; \; -->x-<!--\; -\; -->{\log }_{0.5}^2<!--\; \; -->x-<!--\; -\; -->{\log }_{0.5}^3<!--\; \; -->x-<!--\; -\; -->\cdots <!--\; \cdots \; -->\ge <!--\; \ge \; -->4{\log }_{0.5}<!--\; \; -->x$ Any suggestions how to start?

Is the sum of two odd numbers always odd?

An average mark is computed for 100 students in Business, an average is computed for 300 students in Arts, and an average is computed for 200 students in Science. The average of these three averages is 85%. However, the overall average for the 600 students is 86%. Also, the average for the 300 students in Business and Science is 4 marks higher than the average for the students in Arts. Determine the average for each group of students by solving a system of linear equations.
Let $x_1,x_2,x_3$ represent the students in business, arts and science respectively.

$100x_1+300x_2+200x_3=86$
$100x_1+200x_3+4=300x_2$Not sure as to how to create an equation for the first one.