Questions related to real and complex logarithms.

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2answers
15 views

Different Polynomial Expansions of Natural Logarithm

I was recently Taylor-expanding ln around $(1,0)$. I noticed that this polynomial will have a range of input that converges between $0$ and $2$ regardless of Taylor ...
0
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0answers
6 views

Logistic Scoring Correction

"Consider the logistic curve $f(x)=\frac{1}{1+e^{-bx}}, -1 \leq x \leq 1$. We wish to use this curve to make a scoring correction formula $g(x)$ for an $n$ item test. The domain and range are both ...
0
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0answers
13 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
6
votes
2answers
56 views

Is this summation solvable? $S_n = \sum_{i = 1}^{n}\log_i{(n)}$

Is it possible to solve a summation with a variable base of log? $$ S_n = \sum_{i = 2}^{n}\log_i{(n)} $$ Should I use the derivative of $\log_i{(n)}$?
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1answer
26 views

Evaluate $\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt$

$\lim\limits_{x\to\infty}\frac{1}{\sqrt{x}}\displaystyle\int_1^x\ln(1+\frac{1}{\sqrt{t}})dt=?$ If the limit exists with l'Hopital i get ...
2
votes
2answers
39 views

Integral of $\frac{1}{x^2+1}$ using complex partial fractions.

Is there any way to evaluate the following integral via a complex partial fraction decomposition? $$ \int \dfrac{1}{x^2 + 1} \text{ d}x $$ So far I have: $$ \begin{aligned} \int \dfrac{1}{x^2 + 1} ...
1
vote
1answer
8 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
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2answers
28 views

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?
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3answers
41 views

Evaluate the log expression

Evaulate : $$ \frac{1}{\log_{xy} (xyz)} + \frac{1}{\log_{yz} (xyz)} + \frac{1}{\log_{zx} (xyz)} $$ I think that the following property of log will be used: $$ \log_a (b) * \log_b (c) * log_c (a) ...
0
votes
1answer
22 views

Dynamic Sizing of Circles Along a Logarithmic Spiral

I have created an logarithmic spiral in HTML canvas, and plotted circles along it. Using your mouse scroll wheel you can zoom in and out of the spiral (which works) – but I am having problems updating ...
1
vote
3answers
30 views

Find the largest possible root of a number that is whole

Using 8 as an example radicand, the degree would be 3 because ∜8 is not a whole number, while √8 is not the largest possible whole root. This type of problem is easy to calculate mentally with small ...
1
vote
3answers
69 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
0
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1answer
26 views

reverse a logarithm

I have some data which produces the following logarithmic curve. As you can see, the curve produces the exact opposite of what Im trying to achieve (my data is the line with dots, the logarithm is the ...
0
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4answers
33 views

Solving a logarithmic system of equations

I am working on a test study guide and I can't seem to get the correct answer for this system of equations: \begin{align*} \ln(x) &= 3\ln(y) \\ \ 3^x &= 27^y \end{align*} I'm not ...
-1
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0answers
19 views

logarithm fit, not working against input data

Im trying to do a logarithm fit on the following data, but it just doesnt seem to fit. Is there something Im doing wrong? Im adding the data into this online calculator - ...
0
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2answers
40 views

Integral of $e^x ln(e^{2x} - 4)$

Find the integral from ln4 to ln6 of $$e^x \ln(e^{2x} - 4)$$ I factored $$\ln(e^{2x} - 4)$$ to get $$\ln((e^{x} - 2)(e^{x} + 2))$$ Then I separated this to get: $$e^x\ln(e^{x} - 2) + e^x\ln(e^{x} + ...
0
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2answers
23 views

Approximating the sum of integers with the logarithm

Why does the following hold? $\sum_{j=1}^{n-1}j \to \log(n) \text{ as } n \to \infty$ Thanks!
0
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0answers
20 views

Function that divides expression by unit [on hold]

Originally posted here. This is really a technicality, but something I came to think about recently that it is about time I asked about: Is there an established function which divides an expression ...
2
votes
0answers
16 views

Lipschitz continuity in two variables [duplicate]

Prove that $y \mapsto f(x,y)$ is Lipschitz continuous, where $$f(x,y) = \frac{y}{x} \ln{\frac{y}{x}}, \ \ \ |x-1| \leq \frac{1}{2}, |y-1| \leq \frac{1}{2}e$$ I tried to solve this, but I find it ...
0
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2answers
41 views

Is there a simple algorithm for exponentiating large numbers to large powers?

I've been thinking about this for some days, a multiplication is a lot of sums, so: $$75\times 75=\overbrace{75+75+75+75+75+75+75+75+\cdots}^{\text{75 times}}$$ But then, there is a simple algorithm ...
1
vote
3answers
52 views

Proof that $b^{\log_b(x)} = x$

I understand that the exponential functions are inverses, and would therefore map $x$ when formed as a composition, but I cannot find any formal mathmatical proofs. My thought process is: ...
0
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2answers
22 views

Find the inverse of the function

Find the inverse of the function $f(x) = -2 \cdot4^{2(x-3)} - 1$.
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0answers
11 views

How would go about creating the chart used in the link provided? [closed]

http://www.reddit.com/r/dataisbeautiful/comments/22lrdc/proof_that_dexter_and_how_i_met_your_mother_had This graph seems to normalize data across a number of different TV shows. The finale is ...
1
vote
3answers
25 views

Why $x-\ln (1+e^x)=c$ has a solution for every $c<0$ and not otherwise?

The equation $x-\ln (1+e^x)=c$ has a solution for every $c<0.$ Why is that restriction needed? Why are not we allowed to take positive $c$? I can see that $c=0$ is impossible as $x=\ln ...
2
votes
4answers
69 views

Find all $a$ such that $\lim_{x\to\infty}\left( \frac{x+a}{x-a} \right)^x = e$.

Saw this problem and I thought I'd take a shot at it: Find all $a$ such that $$\lim_{x\mathop\to\infty}\left( \frac{x+a}{x-a} \right)^x = e.$$
2
votes
2answers
44 views

$\lim_{k\rightarrow \infty}\frac{2^k}{\gamma}\log\mathbb{E}[e^{-\gamma \frac{X}{2^k}}]$

I am trying to find a limit for this expression $$\lim_{k\rightarrow \infty}\frac{2^k}{\gamma}\log\mathbb{E}[e^{-\gamma \frac{X}{2^k}}]$$ I have so far found these bounds: ...
0
votes
0answers
23 views

Function inverse mapping [0, +inf) to [0, 1)

I have a measure ($x$) which the domain is $[0, +\infty)$ and measure some sort of variability. I want to create a new measure ($y$) that represents regularity and is inverse related to $x$. It is ...
0
votes
1answer
36 views

If $y=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\dots$ and $z=-y-\frac{y^2}{2}-\frac{y^3}{3}-\dots$ then $\ln (\frac{1}{1-e^x})$

For a nonzero number $x$, if $y=1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\dots$ and $z=-y-\frac{y^2}{2}-\frac{y^3}{3}-\dots$ then the value of $\ln (\frac{1}{1-e^z})$ is ..... I can see that $y=e^{-x}$ ...
2
votes
4answers
173 views

Derive an equation for derivative of ln x

$\frac{d}{dx}e^x = e^x$ use this fact together with the definition of the natural log $\ln x$ as the inverse of the function of $e^x$ to derive an equation for the derivative of $\ln x$.
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2answers
35 views

Working with log numbers [closed]

I am given $$ 3^{\log_{2014} x} = 4^{\log_{2014} 9} $$ and must solve for $x$.
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3answers
15 views

Logarithm Equations: Solving for variable

This is similar to the last question I asked, but I am just unsure about how to work this problem. The equation is $2\ln(x) + 3 = 0$ Please show the steps.
1
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3answers
26 views

Logarithmic equations: Solving for the Variable

The equation is $(3/2)\ln x= -2$. I am not sure how to work this one. If anyone could show all the steps that would be a great help. I tried working it out and got down to $x^{3/2}=e^{-2}$ that is ...
2
votes
4answers
26 views

Logarithmic Equations and solving for the variable

The equation is $\ln{x}+\ln{(x-1)}=\ln{2}$ . I have worked it all the way through, and after factoring the $x^2-1x-2$ I got $x=2$, $x=-1$, but my question is: Can we have both solutions or couldn't we ...
2
votes
0answers
42 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
1
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0answers
56 views

Is this correct: $\ln({(-1)}^{2x-1})=(2x-1)\ln(-1)$?

I would expect the answer to be positive, but it appears otherwise for some values of $x \geq 1$. Here is a simple C++ code that I have used in order to test this: ...
0
votes
1answer
24 views

Logarithmic Equations: Solving for the unknown variable

What is $y$ in $$3^{2y}\cdot3^{\log_{3}(1/3)}=9$$ I apologize if this is confusing, i wasn't sure how to type this equation in here to ask it. If you can please show the steps it would help me ...
0
votes
0answers
22 views

Is this basic proof complete?

I have a problem with proving that the limit as x goes to infinity for lnx/x is 0. Take the most basic approach: Note that the derivative of lnx is 1/x whereas x has a derivative of 1. Hence, lnx is ...
0
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1answer
14 views

Change base log formula?

Im trying to change the base log from ln to log with the following formula. y = a * ln⁡(x+c) + b The ln equation is: ...
3
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1answer
31 views

Show that any invertible matrix has a logarithm.

I was trying to remember how to show that any invertible matrix has a (possibly complex) logarithm. I thought what I came up with was kind of cool, so I thought I'd post my answer here.
0
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3answers
26 views

How do you solve a equation by converting to logarithm form for the problem$3e^{x-4} +2=83$?

$3e^{x-4} +2=83$ I understand converting logarithms but i dont understand how to convert e into log form and solve. help would be deeply appreciated.
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2answers
19 views

Conformal Mapping Between Two Domains (log)

Does anyone have a recommendation as how to go about solving this problem? I want a conformal from G to H where $$ G = \{ z \in \Bbb C \ | \ |z|<1, |z+i|>\sqrt{2} \}, S = \{ z \in \Bbb C \ | \ ...
0
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2answers
30 views

How to substitute $\log_{10}$ for $\ln$ function?

Im wondering how I could go about substituting $\log_{10}$ for $\ln$ in the following formula? $y=a+b\ln(x+c)$ Is there a simple way of doing this? Cheers
0
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2answers
27 views

How can i solve limit.

Hello I have a silly question: How can i show that $\lim\limits_{x \to \infty} \dfrac{\log(x+1)}{\log{x}}=1$. Thank you.
0
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2answers
40 views

A definite integral involve Logarithmic Functions

Here is the integral body: $$\int_0^m {{x^a}\ln \left( {x + b} \right)dx,a > - \frac{3}{2},b > 0} $$
1
vote
3answers
79 views

Convergence of series minus logarithm

im trying to solve this problem since two, three days.. Is there someone who can help me to solve this problem step by step. I really want to understand & solve this! $$ Show\ \exists \ \beta ...
3
votes
0answers
32 views

Exercise concerning logarithms…

I have such a problem: find all the values of real parameter "a", for which the following inequality is true for any "x" that belongs to R. I will show you my solution, and please can you verify ...
1
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0answers
32 views

Finding the $\log$ of a matrix by contour integration

My teacher presented this way of determining the logarithm of a matrix $\Omega$ in class today: $$\log \Omega = \frac1{2\pi i}\oint_{\Gamma} (\zeta I - \Omega)^{-1} \log \zeta \,d\zeta.$$ Does ...
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2answers
40 views

Logarithm problems with different bases

$ \log_a{b} \times \log_b{a} = $ ? $ \log_a{b} + \log_b{a} = \sqrt{29} $ What is $ \log_a{b} - \log_b{a} = $ ? 3. What is b in the following: $$ \log_b{3} + \log_b{11} + \log_b{61} = 1 $$ and ...
1
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1answer
21 views

I can't find second solution to this logarithmic problem!

I kind of got stuck on one step in solving a logarithmic equation. The equation given was: x^3lnx - 4xlnx = 0 My steps so far: x^3lnx - 4xlnx = 0 ln((x^x^3)/(x^4x)) = 0 e^ln((x^x^3)/(x^4x)) = ...
1
vote
2answers
18 views

How do I find the inverse of this exponential function?

$x=-3(3^{-x})+9$ I know the steps up until a certain point. $x=-3(3^{-y})+9$ $x-9=-3(3^{-y})$ $\frac{(x-9)}{-3} = 3^y$ $ln (\frac{x-9}{-3}) = -y * ln 3$ Not sure what to do from here. I know I ...