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How to prove that every bijective rational function (over the field of complex numbers) is a linear fractional transformation?

April Bush 2022-06-17

How do I evaluate this log expression?
Evaluate the expression
$\log_{8} 8^{17}$
I ended up getting $8^{x} = 8^{17}$
I'm guessing I find x, but that's a huge number, and I feel like I'm doing this wrong.

hawatajwizp 2022-06-17

Why do I receive the wrong answer when I try to solve this exponential equation?
So I have the equation: $25^{x} = 5^{x} + 6$
My reasoning is if you make everything to the base 5:
${(5^{2})}^{x} = 5^{x} + 5^{\log_{5} 6}$
Given the bases are the same we can do:
$2 x = x + \log_{5} 6$
$x = \log_{5} 6$
This answer is wrong however, why is this? Once I've the same bases why can't I do this? Furthermore why couldn't I just take logs of both sides of the original equation?( $25^{x} = 5^{x} + 6$ ). What law of logarithms stops me from doing this, why?
Thank you.

tr2os8x 2022-06-17

Limit of $x^{x}$ as x tends to 0
I am trying to solve the following limit:
$lim_{x \to 0} x^{x}$
The only thing that comes to mind is to write $x^{x}$ as $e^{x \ln x}$ and getting the right sided limit would be easy but I don't see how I could get the left sided one seeing that the $\ln$ is not defined for negative numbers.
Is there something I am missing or is there another way to go about it?
P.S.:I don't know anything about derivatives so please keep it to the limits.

Poftethef9t 2022-06-16

graphing $f (x) = x \ln (1 + \frac{1}{x})$
I was assigned to draw the graph of this function $f (x) = x \ln (1 + \frac{1}{x})$
When I calculate $lim_{x \to \infty} f (x)$ but the teacher said it's not correct even though its graph on the internet shows that $lim_{x \to \infty} f (x) = 1$
Please tell me where did I go wrong?

Ayanna Trujillo 2022-06-16

How to solve $lim_{x \to 1} \frac{\ln (x)}{x - 1}$ using pre-derivative calculus?

Kassandra Ross 2022-06-16

Let $f : Q ⟶ Q$ be a continuous function. Is there necessarily an interval $(a, b)$ , with $a, b \in Q$ such that $a < b$ , such that the restriction of $f$ to $(a, b) \cap Q$ is a rational function?
My guess is that the answer is negative. I tried to prove it as follows: I took a countable and dense set ${r_{n} ∣ n \in N}$ of irrational numbers and defined, for each $x \in Q$ , $f (x) = \sum_{r_{n} < x} 2^{- n}$ . This will work, in the sense that $f$ is continuous and that the restriction of $f$ to any interval $(a, b)$ is never a rational function. The problem is that, of course, in general, $f (x) \notin Q$ .

polivijuye 2022-06-16

logarithm equation with different bases.
Why is this like it is? :D
$\frac{1}{\log_{a} e} = \ln (a)$
I'm solving some exercises and I ran up to this? Maybe it's really banal, but please explain me...

hawatajwizp 2022-06-16

Are there any results to determine whether the given power series of real variable converges to a rational function? I mean just analyzing the coefficients of the series. One way is to find the sum function which is not always easy to find.

Misael Matthews 2022-06-15

Precalculus - Exponential and Logarithmic Equations
Mike Kallenberg deposited some money in a bank account that earns 5.6% interest compounded continuously. How long would it take to double the amount in money in Mr. Kallenberg's account?

glycleWogry 2022-06-15

Trying to graph $- \log_{2} x$ , $\log_{2} (- x)$ , and $- \log_{2} (- x)$
I don't understand which one reflects over the x axis, the y axis?

Amber Quinn 2022-06-15

Ln Series Summation
I have been given:
$\ln n = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n} for sufficiently large n$
Which I can equate to $\ln (n) = \sum_{i = 1}^{n} \frac{1}{i}$
The series I need to sum is:
$\frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{79999}$
I know this can be determined by doing $\ln (80000) - \frac{1}{2} \ln (40000)$ , but is it valid to say:
$\ln 2 n - 1 = \sum_{i = 1}^{n} \frac{1}{2 i - 1} = \frac{1}{1} + \frac{1}{3} + \frac{1}{5} + \dots + \frac{1}{2 n - 1} ?$
If not, a brief reason why it isn't valid would be appreciated!
Thanks in advance
PS. This is part of a question from the Cambridge maths paper STEP III 2008

Finley Mckinney 2022-06-15

Proof of generalization of a particular limit converging to $e^{\frac{1}{(p - 1)^{2}}}$
I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th one . Prove that for all $p \in N$
$lim_{n \to \infty} \frac{(1^{1^{p}} \times 2^{2^{p}} \times 3^{3^{p}} \times \dots n^{n^{p}})}{n^{\frac{1}{p + 1}}} = e^{\frac{1}{(p - 1)^{2}}}$
I tried to manupilate this with the properties of logarithms but failed.

telegrafyx 2022-06-15

Evaluate $\frac{d}{d x} {(\sin x)^{\cos x} + (\cos x)^{\sin x}}$ with logarithmic differentiation
Spivak asks us to evaluate
$\frac{d}{d x} {(\sin x)^{\cos x} + (\cos x)^{\sin x}}$
by logarithmic differentiation. Does he mean for us to evaluate each term separately (which seems to turn out to be cumbersome), or is there something else I'm missing?

arridsd9 2022-06-15

How to show that the two rational functions $a (t), b (t) \in K (t)$ when $Char (K) \neq 2$ are algebraically dependent.
(there is polynomial $P (x, y) \in K [x, y]$ such that $P (a (t), b (t)) = 0$ )?

Petrovcic2x 2022-06-15

Solving a logarithmic polynomial
I want to solve this equation for $x$ : $\frac{1}{\sqrt{2 π x}} {(\frac{e z}{2 x})}^{x} = ϵ$
Is there a closed form for it, or does it have to be solved numerically? I can turn it in to this:
$2 x \log (e z) - (2 x + 1) \log (2 x) = \log (ϵ^{2} π)$
Which is in a form similar to $a + b \log x + c n + d n \log n = 0$ , which is a deceptively simple equation that I don't know how to solve analytically. It feels very similar to a polynomial, but instead of powers of $x$ it's increasing $O (x)$
Failing a closed form answer, is there a useful closed form bound?

Amber Quinn 2022-06-15

How to solve the inequality $n! \leq n^{n - 2}$ ?

Micaela Simon 2022-06-15

Comparing logarithmic functions. Master Method
I'm learning the master method and am looking for help on how to best approach comparing two functions asymptotically. More specifically, I have:
T(n) = 3T(n/5) + lg^2(n)
and so by the Master method I am comparing
n^(log_5(3)) with lg^2(n)
I tried graphing the two functions and it looks like lg^2(n) is larger. But the solution says otherwise. (ie. Case 1.) Can anyone help clear the fog for me? Thanks.

Dania Mueller 2022-06-15

Confusion between negative and positive signs in natural logarithm
If $z = - 0.1887 \cdot (x^{0.7637}) \cdot (y^{0.2306})$
Its natural logarithm will be
$\ln (z) = - [\ln (0.1887) + 0.7637 \ln (x) + 0.2306 \ln (y)]$
or
$\ln (z) = - [\ln (0.1887) - 0.7637 \ln (x) - 0.2306 \ln (y)]$ ?
Thank you in advance

enrotlavaec 2022-06-15

Using this rule: Integral Calculus "Rational Function Rule"
$\int \frac{1}{a^{2} - x^{2}} d x = \frac{1}{2 a} \log_{e} (\frac{a + x}{a - x}) + c$
I have integrated a function, but my question is if I'm solving for $x$ , in the next line I must raise both sides to the base of $e$ , and there is still $+ c$ (constant) on the end of the integrated function, so do I leave it as plus $c$ on the RHS on the equation (not do anything to it, so when I want to work it out by substituting $(x_{1}, y_{1})$ at the end it will be $x_{1} = \dots y_{1} \dots + c$ ) or when I raise both sides to the base of $e$ does $c$ , the constant become $c * e \dots$ instead of $+ c$ ?
Any help would be great.

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