Questions related to real and complex logarithms.

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1answer
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Given $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$and smallest $b$ such that $a < \log_{10} 2 < b$

If one uses only the information $10^3=1000,10^4=10000,2^{10}=1024,2^{11}=2048,2^{12}=4096,2^{13}=8192$,what are the largest $a$ and smallest $b$ such that one can prove $a < \log_{10} 2 < b$ ...
0
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1answer
47 views

Domain of the nested logarithmic function

Find the domain of the definition of the function $$f(x)=\log_{0.3}\left(\log_{0.5}\left(\log_{0.8}\left(x^2-x+1\right)\right)\right)$$ My Try: I assumed $$f_1(x)=x^2-x+1$$ $$f_2(x)=\log_{0.8}(...
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0answers
59 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
3
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3answers
86 views

How to compute $\int_{0}^{(e-1)^2}{\ln(\sqrt{x}+1)} \,\mathrm dx $?

I have a problem with this integral. $$\int\limits_{0}^{(e-1)^2}\!\! \left({\ln(\sqrt{x}+1)} \right)\,\mathrm dx $$ I applied the substitution method $t = \sqrt{x}+1$, $2t = dx$ I changed integration ...
2
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3answers
80 views

For which values of $a\in\mathbb{R} $ the equation $2 \log(x+3)=\log(ax)$ has exactly one root?

I have to investigate the possible roots of the equation according to $a$, i.e. i have to see whether there is only one root, two roots, or no roots and also what their sign is each time. This is from ...
2
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3answers
40 views

Divergence of $\sum_{n\geq 2} \frac{1}{\ln^p n}$ for $1<p\leq \infty$ [closed]

Can anyone help me to prove that $(x_n)\notin l_p$ with $x_n=\frac{1}{\ln^p n}$? Suppose $1< p<\infty$.
1
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1answer
33 views

How to derive this simple equality?

Let us define $L_i\triangleq \log \left( \dfrac{Prob(x_i=+1) }{ Prob(x_i=-1)} \right)$ $E\{x_i\} \triangleq Prob(x_i=+1)-Prob(x_i=-1)$ I need to show that \begin{equation} E\{x_i\} = \tanh(L_i/...
8
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3answers
246 views

Integral $\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$

It's a follow-up to my previous question. Can we find an anti-derivative $$\int\arcsin x\cdot\ln^3x\,dx$$ or, at least, evaluate the definite integral $$\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$$ in a ...
10
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4answers
259 views

Evaluating $\int_{0}^{\pi}\ln (1+\cos x)\, dx$ [duplicate]

The problem is $$\int_{0}^{\pi}\ln (1+\cos x)\ dx$$ What I tried was using standard limit formulas like changing $x$ to $\pi - x$ and I also tried integration by parts on it to no avail. Please help....
0
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2answers
38 views

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$.

If $f(x)=\log \left(\cfrac{1+x}{1-x}\right)$ for $-1 < x < 1$,then find $f \left(\cfrac{3x+x^3}{1+3x^2}\right)$ in terms of $f(x)$. My Attempt $$f \left(\cfrac{3x+x^3}{1+3x^2}\right)=\log\...
2
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1answer
82 views

$c= (a^{x}-b^{x})$ where $a$,$b$ and $c$ are known real constants. Solve for $x$.

I tried taking $\log$ on both side but i ended with $\log(a^{x}-b^{x})$ which is difficult to solve. Does anybody has idea how to solve the above equation for $x$.
1
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0answers
30 views

Problem with custom made natural log and power functions

I have made these two functions with the help of posts on math.stackexchange.com. For ln I'm using information gathered from Calculate Logarithms by Hand and for ...
3
votes
2answers
96 views

How can a positive integrand integrate to 0? [duplicate]

I integrated $\dfrac{\log x}{1+x^2}$ from $0$ to infinity with residue calculus and got... $0$. This also agrees with Wolfram Alpha. How can this be? Is it due to the behavior of $\log(x)$ near ...
1
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1answer
28 views

Hyperbolic Cosine: System of Equations, Isolate Variables

Background information that may help you answer: Alright, so I'm working on a formula that posits that there are a unique pair of coordinates $(x_1, y_1)$ and $(x_2, y_2) $ on the hyperbolic cosine ...
1
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1answer
80 views

Growth rates slower than logarithmic? [closed]

So far, I've been able to determine growth rates using the following limit:$$\lim_{x\to\infty}\frac{f(x)}{g(x)}$$Which, if need be, can be solved with calculus. From this, I deduced that it is very ...
9
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11answers
472 views

How to prove that $\log(x)<x$ when $x>1$?

It's very basic but I'm having trouble to find a way to prove this inequality $\log(x)<x$ when $x>1$ ($\log(x)$ is the natural logarithm) I can think about the two graphs but I can't find ...
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3answers
93 views

What am I missing in this argument for $\lim\limits_{x\rightarrow \infty} \ln x = \infty$?

In an appendix of Stewart's Calculus, the logarithmic and exponential functions are built up starting from the defnition $\ln x = \int_1^x \frac{1}{t}\,dt$. Having shown that $\ln(x^n) = n \ln(x)$ ...
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4answers
538 views

Yet another log-sin integral ${\large\int}_0^{\pi/3}\log(1+\sin x)\log(1-\sin x)\,dx$

There has been much interest to various log-trig integrals on this site (e.g. see [1][2][3][4][5][6][7][8][9]). Here is another one I'm trying to solve: $$\int_0^{\pi/3}\log(1+\sin x)\log(1-\sin x)\,...
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4answers
142 views

What is the $n^{th}$ derivative of $\log_x(e)$?

I happened to stumble upon the integral of $\log_x(e)$, finding it to apparently be non-elementary. So I had to see if I could discern a pattern by differentiating, much like finding the integral of $...
2
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3answers
42 views

Solving logarithm problem, Found the value

This is my first question in this place, I don't know how to solve my problem. I have this equation, I need to find the central value based on this equation: $k = 0.2$ $$k = 2^{(\frac{1}{24} + \...
0
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5answers
105 views

Can't solve this exponential equation: $5^{x+1}-3\cdot 5^{x-1} - 6\cdot 5^x+10 = 0$ [closed]

How does one solve for $x$ in the following: $$5^{x+1}-3\cdot 5^{x-1} - 6\cdot 5^x+10 = 0$$
4
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5answers
158 views

Complex Number in a logarithm

I am having a little trouble understanding complex numbers with logarithms. How would I do the log of $e$ ($\log_{i}{e}$)? What I did firstly was to do $\frac{\log{e}}{\log{i}}$. I don’t have any idea ...
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4answers
59 views

What will be the value of n?

$$ \log_{10}2 = \frac{1}{n} \times \log_{10}n\, $$ Can anybody please help me finding the value of n ? Sorry for earlier typo..Edited the questions
6
votes
2answers
158 views

Does logging infinitely converge?

Trying to evaluate $$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$For some fixed $x$ produces a complex answer that appears to converge, at least sometimes. So I want a proof that this converges for ...
1
vote
1answer
91 views

Limit to infinity and infinite logarithms?

When trying to evaluate$$\ln(\ln(\ln(\ln(\cdots\ln(x)\cdots))))$$I noticed that the answer was bound to be complex for any $x$. Plugging in a very, very large real number in for $x$ will eventually ...
8
votes
3answers
224 views

Integral $\int_0^{1/2}\arcsin x\cdot\ln^2x\,dx$

I'm interested in this integral $$\int_0^{1/2}\arcsin x\cdot\ln^2x\,dx$$ My idea was to first evaluate $$\int_0^{1/2}\arcsin x\cdot x^a\,dx=\frac{2^{-a}\,\pi-6B_{1/4}\left(\frac{a}{2}+1,\frac{1}{2}\...
3
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0answers
81 views

How prove that $\log_3{5} + \log_2{5}$ is irrational? [duplicate]

How prove that: $\log_3{5} + \log_2{5}$ is irrational? $$\log _3{5} + \log _2{5}=\frac{1}{\log _5{3}}+ \frac{1}{\log _5{2}}= \frac{\log _5\left( 3\cdot 2\right) }{\log _53 \cdot \log _52}$$
2
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1answer
38 views

Approximating logarithmic series

I'm trying to solve the below recurrence $$T(n) = 2 T(n/2) + n/\log n$$ I referenced some online articles they all are approximating $n \sum_{i=0}^{h-1} \frac{1}{\log n - i}$ to $n \sum_{i=1}^{h-...
0
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1answer
38 views

converting the sum of numbers to logarithms

If I have three number, $[0.2, 0.3, 0.4]$ then I can take the sum simply by adding the term: $0.2+0.3+0.4=0.9$. The proportion of the first element is then $0.2/0.9 = 2/9$. Now I don't know what the ...
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3answers
63 views

Why can't you add a coefficient before the logarithm base change rule?

Why is it that the rule $$ \log_b(x) \ = \ \frac{\log_c(x)}{\log_c(b)} \ $$ (the logarithm base change rule) is true but $$ \ a \log_b(x) \ = \ \frac{a \log_c(x) }{ a \log_c(b) } \ $$ isn't? For ...
2
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2answers
142 views

Is $0 \times \ln(0) =\ln(1) $ true?

Can we affirm that: $0 \times \ln(0) = \ln(0^0) = \ln(1) = 0$? The problem is $\ln(0)$ is supposed to be undefined but it works
0
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1answer
39 views

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 ...
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2answers
34 views

Can the complex square root of $z\sin z$ be defined in a neighborhood of the origin? (I.e., including the origin)

Edit: on a second thought, I don't think it's possible since $$ f(z) = \sqrt {z\sin z} = e^{\large \frac{1}{2} \log z}e^{\large \frac{1}{2} \log\sin z}$$ $$e^{\large \frac{1}{2} (\ln|z| + iArg(z))}e^...
23
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2answers
425 views

Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of Malmsten—Vardi&...
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1answer
34 views

Determining the value of parameters given constraints

If $$\frac{x(y+z-x)}{\log x}=\frac{y(z+x-y)}{\log y}=\frac{z(x+y-z)}{\log z}$$ and $$ax^yy^x=by^zz^y=cz^xz^y$$ then what is the value of $a + \frac b c$? I am getting as $ax^yy^x=by^zz^y=cz^xx^z$ ...
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votes
2answers
110 views

Prove $f(x)=2^x$ is continuous

I have show that $f(x)=2^x$ is continous by using the Weierstrass definition (epsilon-delta). I set apart two cases. The first one $x>x_0$ was good. The second one is the problem right now. $|f(x)-...
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3answers
86 views

The domain of $\ln(x)^{\ln(x)}$

I'm a little bit confused! what is the domain of this function: $$ \ln(x) ^{ \ln(x) } $$ this function, in fact, is: $$ \exp(\ln(\ln(x))\cdot\ln(x)) $$ so the domain would be: $$ x>1 $$ But: $...
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2answers
40 views

How to use the properties of the logarithmic function

I'm coding the game asteroids. I want to make a levels manager who can create a infinity number of level increasing in difficulty. My levels have as parameters : The number of asteroids on the ...
7
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2answers
147 views

Integral $\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$

I need to evaluate this integral: $$\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$$ Apparently, Maple and Mathematica cannot do anything with it, but I saw similar integrals to be evaluated in ...
5
votes
4answers
103 views

Can you compute $\int_0^1\frac{\log(x)\log(1-x)}{x}dx$ more precisely than $1.20206$ and do a comparision with $\zeta(3)$?

I know from Wolfram Alpha that $$\int_0^1\frac{\log(x)\log(1-x)}{x}dx=1.20206$$ and in the other hand, too from this online tool that $$\int\frac{\log(x)\log(1-x)}{x}dx=\mathrm{Li}_3(x)-\mathrm{Li}_2(...
0
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3answers
46 views

Log transformation

Supose I have a series of numbers from 1 to 10. Their mean value is 5.5. Now supose I apply some transformation like $y=2x+1$. Now their mean value is 12. Now, if I want to get back the original mean, ...
4
votes
1answer
127 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
2
votes
2answers
84 views

How to solve this equation algebraically?

I've come across this interesting equation which I do not know how to solve. The equation is: $$e^x+\log x =1$$ I used WolframAlpha to solve it and it got but, it didn't provide any solutions. The ...
7
votes
1answer
124 views

Is $a^{\ln b} = b^{\ln a}$?

I was struggling with a math problem, namely, a limit with a power to the log of something. While I was struggling with it, I found out that $$a^{\ln b} = b^{\ln a}$$ for all positive values that I've ...
0
votes
2answers
26 views

Question about expressing logarithms

If logb (a) = m and logy (b) = c, then find loga (y) in terms of variable c and m This is what I have so far logb (a) = log a / log b logy (b) = log b / log y =(log a / log b)(log b / log y) = ...
0
votes
3answers
76 views

Logarithm of a transcendental number

Can anything be said about the nature of the number $\log y $ where $y $ is a transcendental number not of the form $y=e^x $ or written trivially in that form using $x=\log w $ for some $w $ ...
3
votes
2answers
108 views

Can $\log(x)\log(y)$ be reduced?

I'm currently taking Pre-Calc and am learning about logs. I know that $\log(xy) = \log(x) + \log(y)$, but can $\log(x)\log(y)$ be reduced further?
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0answers
16 views

Multiplicative & additive measurement error models concerning logarithms

I understand that taking the logarithm of the multiplicative error model transforms it into the additive error model. Let $y'$ be the observed response variable, with $y$ being the true response ...
1
vote
2answers
60 views

If $\ln x$ is integrable, then is $x \ln x$ also integrable?

I have a very simple problem. Assume we have a finite measure $\mu$ on $[1,\infty)$, and \begin{align} \int_1^\infty t ~d\mu(t) < \infty. \end{align} My question is if this implies \begin{align} \...
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2answers
39 views

Simplifying a log function

How come: y=c1*e^(c2*t) is simplified to: ln(y)=ln(c1)+c2*t ? What I got is: ...