Questions related to real and complex logarithms.

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How to compute the logarithm of a computable number

Let's say you have a computable number $x>0$. $\ln(x)$ is computable as well. Given the computability of $x$, what is a computation for $\ln(x)$. I am using the definition where $a$ is computable ...
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2answers
43 views

Simplifying logarithm question

Without worrying about the background, I have a question that asks to solve for n. Pardon my formatting, but it seems understandable this way for the time being until I edit it: $$4n^2 = 256 ...
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0answers
7 views

negative sign in direction of wave propagation

Say I have a EM wave that goes in the Z direction and E=Eo*exp(-jkz). Why does the negative sign mean the wave travels in the +Z direction and exp(+jkz) means it travels in the -Z direction?
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2answers
46 views

The minimum value of $\log_{10}x+\log_x 10$

Notation: $\log:=\log_{10}$ $\log x+\log_x 10$ $=\log x+ \frac{1}{\log x}$ $=\log(x \cdot \frac{1}{x})$ $=\log 1$ $=0$ Is the process correct? I doubt this is wrong. Please help. ...
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3answers
68 views

logarithmic and polynomial equation

I have the following $(1-a^x)/x=b$ Can this be solved for x ? (if yes, how, if not why) I have gotten to many forms, but can't seem to isolate x.
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4answers
55 views

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through ...
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1answer
37 views

Difference between the formula of Roger Cotes and Euler

What was the difference between the formula that Roger cotes derived and that euler got? I mean to say that Euler got the following formula : $$e^{ix} = \cos x+i \sin x$$ And Cotes got the following ...
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2answers
34 views

A Simple Logarithm Question

Solve for $x$: $\log_2 (2x+8)=3$ Correct Solution: $2x+8=2^3$ $2x+8=8$ $2x=0$ $x=0$ Why doesn't this work: $\log_2 (2x+8)=3$ Expand: $\log_2(2x)+\log_28=3$ $\log_2(2x)+3=3$ $\log_2(2x)=0$ ...
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2answers
63 views

Is $\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$? [closed]

Is $$\sqrt{\log (n)}=\frac{1}{\sqrt{2}}*(\log n)$$
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3answers
81 views

How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
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1answer
40 views

Solve by separation of variables: $\frac{dx}{dy}y\ln|x| = \big(\frac{y+1}{x}\big)^2$

I need to solve the problem above using separation of variables. I got as far as the below but it seems too complex to be right. Am I wrong somewhere? Because I think my final answer needs to simplify ...
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3answers
84 views

Find sum of series [closed]

I need to find the sum of the following series: $$\sum_{n=2}^\infty \ln\left(1-\frac 1{n^2}\right)$$ How to proceed with this?
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2answers
31 views

Sequences identity

I have some problems to find a way to prove the following statement, if someone could give me any suggestions would be grateful: Show that $$ log\text{(}a_{n}+\text{1})\approx a_{n} $$ when $$ ...
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2answers
251 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to an algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
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2answers
64 views

Evaluate $\log_{2005}(1/2)\log_{2004}(1/3)\log_{2003}(1/4)\ldots\log_2(1/2005)$ [closed]

The numbers 2005, 2004, 2003, ..., 2 are the bases. I cannot understand how to start the question. Please help. What to do in these type of questions? Thanks in advance.
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2answers
21 views

Logarithm where $0<a<\frac{1}{2}$. Find $x$

Given that $\log_a(3x-4a)+\log_a(3x)=\frac{2}{\log_2a}+\log_a(1-2a)$ where $0<a<\frac{1}{2}$. find the value of $x$. I got the attempt until $x=\frac{2(a+\sqrt{(a-1)^2}}{3}$ and ...
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1answer
50 views

Are these proofs of the 1st and 3rd Laws of Logarithms valid?

Disclaimer: I dont mean that I've discovered a conceptually completely different way of proving those laws, of course. I just found myself proving them like this and then realized that they're ...
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1answer
53 views

For which values of a parameter an equation has one Real root

The following equation is given $$\log_{x-1}(x^2+2ax) - \log_{x-1}(8x-6a-3)=0$$ And I am trying to find for which values of $a$ it has only one root, which is real. It is obvious that $$x-1>0 ...
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0answers
36 views

How do I show that the integral $\int_0^\infty x^{-a} |\log x|^b dx$ only converges when $a = 1$ and $-2 < b < -1$?

This came up in a previous question, but was closed because the question wasn't terribly clear. I don't want to edit the other question substantially because it's not mine so I'm asking a new one and ...
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5answers
54 views

Solving logarithmic equations including x

Let $$\log_3(x-2) = 6 - x$$ It's obvious drawing the graphs of the two functions that the only solution is $x=5$. But this is not really a proof, rather than observation. How do you prove it ...
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2answers
57 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
2
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3answers
42 views

Logs rules and Solving

I've got the equation : $$-1=\frac{-8e^{-t} + 3e^t}{2e^t}$$ I've moved some stuff around to get : $5e^t = 8e^{-t} $ But not sure where to go from here. Thanks for any help
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3answers
42 views

Set of real $a$ so that the inequality is defined but isn't true for a real $x$

$$x(x-\sqrt {4+\log_a7})\lt \log_7 \frac a{49}$$ I reach the interval $(0,1)$ after looking for the discriminant of the quadratic to be less than zero. However, the solution in the book is an ...
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1answer
48 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
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1answer
19 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
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1answer
59 views

Derivative of $(\ln x)^e$ [duplicate]

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ...
2
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1answer
33 views

Confused with integral and natural logarithm

When reading about ideal gas and adiabatic expansion, I got stuck with the following: $$W_{ab}=\int_{{\it V_a}}^{{\it V_b}}\!\,{\rm ...
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2answers
50 views

How to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$

I want to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ if $p > 1$ and $r \in \mathbb{N}$. To show this, I wanted to use that $\lim_{x \to 0} x \ln x = 0$, and in fact if $p \geq r$ we can write ...
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1answer
15 views

Integer solutions to the inequality $\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$

$$\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$$ If $x$ is of the interval $[-8,10]$ Now I solved this, tried to limit $x$ as much as I could but I consistently get that there should be $10$ values of $x$ ...
2
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1answer
74 views

Solve for $x$: $x =\ln(x)^4$

I plotted the functions on both sides and it shows the equations has at least three solutions. Is there some non-interative (not sure if i used this term correctly - i mean the way you would solve, ...
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1answer
17 views

Question about a simple rule of the complex logarithm

According to the Wikipedia page on complex logarithms: Also, the identity $\log(xy) = \log x + \log y$ can fail: the two sides can differ by an integer multiple of $2\pi i$. Does the same hold ...
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1answer
47 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
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1answer
43 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
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0answers
31 views

What properties does $a(n)$ have to fullfill to get $\log(2^{cn}+a(n))\sim\log(2^{cn})$?

Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...). Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, ...
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1answer
36 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
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2answers
26 views

Number of digits

I've been trying to find a solution to this problem for a while but I just can't seem to find the connection between the numbers and I really need help. I apologize if a problem like this one has ...
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1answer
13 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
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25 views

Multiple polylogarithms

$1)$ If $$I(a_o;a_1 \dots a_n ; a_{n+1}) = \int_{a_o}^{a_{n+1}} \frac{dt}{t-a_n}I(a_o;a_1\dots a_{n-1};t)$$ and $G(a_1 \dots a_n;z) = I(0 ; a_n \dots a_1;z)$, where $G$ is a multiple polylogarithm, ...
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3answers
204 views

How to find the inverse of $n\log n$?

So I'm on chapter $1$ of introduction to algorithms & at the end the book proposes a problem: here The answers are there & I was able to work through most of them myself despite my lack of ...
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1answer
16 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
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1answer
35 views

Polylogarithms and the shuffle algebra

$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable changes of variables. Attempt: The di-log is defined as ...
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2answers
43 views

Logarithm with nth root [closed]

I made it but the result is very strange. I want every step to the result $$ \large 6\log_{10}\frac{\sqrt2}{\sqrt[3]{3+\sqrt5}} $$
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4answers
493 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
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1answer
27 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
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1answer
98 views

Are logarithms radicals? [closed]

Does the set of all logarithms with a radical base and argument belong to the set of all radicals? A simple yes, no answer will suffice, an explanation would be wonderful. EDIT 1 Can a logarithm with ...
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2answers
83 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
2
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3answers
53 views

Values of a for which equation $\log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert$ has a unique solution

\begin{equation*} \log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert. \end{equation*} I don't even know how to approach this one, any hints would be amazing. I tried separating into two cases, where ...
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1answer
29 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
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2answers
60 views

Solutions of $2^x 7^{1/x}\le 14$

The solution is supposed to be $(-\infty,0)$ and $[1,\log_2 7]$. What I get when solving the problem is $(-\infty, \log_2 7]$. Where did I get it wrong? I start by dividing both sides by 14, then ...
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1answer
50 views

logarithmic Series

I'm aware that by properties of logarithm $$\sum_{k=1}^n \ln (k) = \ln (n!)$$ My question is if $$\sum_{k=1}^n \ln^2 (k) = \ln^2 (n!)?$$ Because when I am verifying the value where $n = 5$, I get ...