Questions related to real and complex logarithms.

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0
votes
1answer
26 views

Diffie hellman and the discrete algorithm problem

Suppose Alice and Bob are exchanging keys using Diffie-Hellman Key-Exchange Algorithm. a - Alice secret key g - generator p - prime x - the public key passed from Alice to Bob. Eve is listening to ...
25
votes
2answers
1k views

Integral $\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$

Could you suggest any ideas how to evaluate this integral? Is there a closed-form result? $$\int_0^\infty\frac{\ln\left(1+x+\sqrt{x^2+2\,x}\right)\,\ln\left(1+\sqrt{x^2+2\,x+2}\right)}{x^2+2x+1}dx$$
3
votes
1answer
96 views

How do I find the decimal part of the answer of a logarithm problem

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. I have been trying to solve logarithm problems with decimal answer. For example ...
4
votes
4answers
140 views

$\lim_{n \to \infty} n\sqrt 2\, \big(\sqrt{\ln(n+1)}-\sqrt{\ln n}\big) = 0$

I'm trying to prove that $$ \lim_{n \to \infty} n\sqrt{2}\,\left(\sqrt{\ln(n+1)}-\sqrt{\ln n}\right) = 0 $$ But I haven't any ideas how to do it... My calculations shows that this sequence is ...
13
votes
5answers
1k views

Is there any nonconstant function that grows (at infinity) slower than all iterations of the (natural) logarithm?

Is there any nonconstant function that grows at infinity slower than all iterations of the (natural) logarithm?
3
votes
2answers
71 views

Is there a proof that $n^xm^x = (n^x)^{(\log(mn)/\log(n))}$?

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. So the other day I was playing with my calculator and I noticed that $$ 2^x10^x = ...
0
votes
2answers
93 views

Limit of a harmonic subseries minus “its” logarithm

$\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{3k-1} - \frac{1}{3}\ln(n)$ I think that inserting the other terms and then subtracting them would not help. I need just the ideea. Thank you.
1
vote
2answers
112 views

Asymptotics of logarithms of functions

If I know that $\lim\limits_{x\to \infty} \dfrac{f(x)}{g(x)}=1$, does it follow that $\lim\limits_{x\to\infty} \dfrac{\log f(x)}{\log g(x)}=1$ as well? I see that this definitely doesn't hold for ...
0
votes
1answer
34 views

Equations with exponents

I can't remember how to solve equations that have exponent and a variable in them. This is somewhat embarrassing, because this used to be really easy for me. I know that logarithms are involved I just ...
3
votes
1answer
1k views

What is the opposite of logarithmic scale?

If one wants to plot a variable whose value changes drastically, it makes sense to plot its order of magnitude. That way, the smaller values in the plot aren't squished all the way at the bottom. This ...
3
votes
1answer
112 views

Solve for: $8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$

Solve for: $$8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$$ My try: ...
3
votes
1answer
152 views

Solve for: $2\log_3\left(x^2-4\right)+3\sqrt{\log_3\left(x+2\right)^2}-\log_3\left(x-2\right)^2\leq4$

Solve for: $$2\log_3\left(x^2-4\right)+3\sqrt{\log_3\left(x+2\right)^2}-\log_3\left(x-2\right)^2\leq4$$ My try: ...
3
votes
1answer
65 views

Simplifying this logarithm series

$$\sum_{i\; =\; 2}^{99}{\frac{1}{\log _{i}\left( 99! \right)}}$$ How would you evaluate (or at least simplify) this logarithm series?
3
votes
0answers
99 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
2
votes
1answer
88 views

Exact Solution for Logarithmic Equation?

I am faced with this equation, and I don't really know where to begin: $$x^2e^2 - 2e^x = 0.$$ I usually start these types of problems by factoring out a common term, but I don't see any in this ...
0
votes
3answers
86 views

Solving $\log(x+2) - \log(x) = 3$

I have work through the whole problem, but I cannot get passed the last step. The original equation was: $\log(x+2) - \log(x) = 3$ I worked it out to this: $\frac{x+2}{x} = 1000$. I know the answer ...
1
vote
8answers
368 views

Given $2^{x}=129$, why is it that I can use the natural logarithm to find $x$?

I've looked at an example in my textbook, it is: $2^{x}=129$ $\ln \left( 2^{x}\right) =\ln \left( 129\right) $ $x\ln \left( 2\right) =\ln \left( 129\right) $ $ x=\dfrac {\ln \left( 129\right) ...
39
votes
4answers
2k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}\mathrm dx.$$ I do not have a strong reason to be sure it exists, but I ...
0
votes
1answer
50 views

function symmetric around a point

I need some quick help solving this: What is y(ln(2))if the function y satisfies $$\frac{dy}{dx}=1-y^2$$ and is symmetric about the point (ln(4),0)? I know that a function is symmetric about the ...
2
votes
2answers
83 views

Question that can not be solve analytically .

You can know that the solution of this non-linear simultaneous equations is y=2 and x=3; but the question is : How can mathematically ( algebraically ) find this. \begin{array}{lcl} x^y & = & ...
4
votes
1answer
337 views

log(log(123456789101112131415…)))

How would you fin the integer closest to log(log(1234567891011121314...2013)) where the number is the concatenation of numbers 1 through 2013 inclusive. log() in this case is log base 10. Also, how ...
2
votes
3answers
157 views

How to show that $\log (\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2})= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$

If $\log (a +b +c) =\log a+\log b+\log c$ then show that $$\log \left(\frac{2a}{1-a^2}+\frac{2b}{1-b^2}+\frac{2c}{1-c^2}\right)= \log\frac{2a}{1-a^2}+ \log \frac{2b}{1-b^2}+ \log \frac{2c}{1-c^2}$$ ...
0
votes
2answers
31 views

Inequality for negative logarithms?

Given $0 < x < y < 1$, is it possible to prove the following result: $\frac {ln\:x}{ln\:y} < 1$? Thanks
29
votes
2answers
2k views

Closed form for $\int_0^1\sqrt{\frac{2-x}{(1-x)\,x}}\,\log\left(\frac{(2-x)\,x}{1-x}\right)dx$

This is somewhat similar to my previous question: Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$ Is it possible to find a closed form ...
0
votes
1answer
201 views

Logic: Prove Log(base 9) 15 is irrational

Im having trouble with the following proof... Ill post what I have completed so far.. Prove $\log_915$ is irrational. Ill attempt by contradiction assuming $\log_915$ is rational. So, $\log_915 = ...
1
vote
2answers
131 views

Confusion regarding the Logarithmic function change of base formula

My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated. It starts with: $b^{x ...
28
votes
2answers
474 views

Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$

I need to evaluate this integral: $$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$ I tried it in Mathematica, but it was not able to find a closed ...
1
vote
1answer
34 views

Logarithm question equation

I'm stuck on an equation : $$(\log_8 x)^2+2\log_8 x+1=0$$ I've played with it without any success. Any indications would be greatly appreciated... Thank you!
20
votes
2answers
392 views

Integral $\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx$

I need to evaluate the following integral: $$\int_0^1\frac{\ln x}{x^2+1}\cdot\ln\left(\frac{3\,x^2+1}{x^2+3}\right)dx.$$ Could you suggest how to find a closed form for it? I am not sure if there is ...
0
votes
1answer
30 views

logarithms question

I would like to know something about logarithms : If I have, as an example : $(\log_2(x))^2$ How would you simplify that ?? Is it the same as log2(x)^2 Thank you !
0
votes
3answers
48 views

Removing logs from equation

I have a simple question that I need clarification on: If $$\log(a) = \log(b) + c$$ is it true that $$a = b + \exp(c)$$ Is this correct or am I missing something really basic that I cant ...
1
vote
3answers
185 views

Does $\sum_{n=2}^\infty (n\ln n)^{-1}$ diverge?

Is $\sum_{n=2}^\infty (n\ln n)^{-1}=\infty$ ? This seems like elementary calculus, but I can't figure this out. Can anyone supply a hint?
27
votes
3answers
674 views

Integral $\int_0^1\frac{\ln x}{\left(1+x\right)\left(1+x^{-\left(2+\sqrt3\right)}\right)}dx$

There is a curious known integral: $$\int_0^1\frac{\ln\left(1+x^{2+\sqrt{3\vphantom{\large3}}}\right)}{1+x}dx=\frac{\pi^2}{12}\left(1-\sqrt{3\vphantom{\large3}}\right)+\ln ...
0
votes
3answers
62 views

What is the value of $ (a+b) $ where $ a\log_{1971}3 + b\log_{1971}{73} = 2012 $

I have two integers which are a and b . They satisfy the following equation which is $ a\log_{1971}3 + b\log_{1971}{73} = 2012 $ . I want to know the value of $ (a+b) $ . I have tried to solve ...
1
vote
1answer
90 views

How to prove the following inequality of logarithm?

Let $x,y,z\in\mathbb{C}.$ Suppose $$z=\frac{1}{2}(xy\pm\sqrt{x^2y^2-4(x^2+y^2)} ).$$ Show that $$log^+|z|\leq log^+|x|+log^+|y|+log 2.$$ Where $log^+\phi=max\{0,log\phi\}.$ Here we are also ...
0
votes
3answers
424 views

Why is $-\ln(\cos(x))$ equal to $\ln(\sec(x))$?

Why does the value $-\ln(|\cos(x)|)$ become $\ln(|\sec(x)|)?$ I was doing an integral and I got my final answer as that, but I don't understand how you can just send the negative sign inside and make ...
0
votes
1answer
48 views

Principal Logarithmic Question

Here is a question that is driving me insane: Show that $p.v \sqrt{z-1}\times p.v\sqrt{z+1}=-p.v.\sqrt{z^2 -1}$ for $Re(z)<-1.$(p.v. stands for the principal singular valued logarithmic ...
0
votes
2answers
130 views

Use a graph to estimate the time at which the number was increasing most rapidly

For the period from 2000 to 2008, the percentage of households in a certain country with at least one DVD player has been modeled by the function $f(t) = \frac{87.5}{1 + 17.1e^{−0.91t}}$ where the ...
0
votes
1answer
739 views

How should I express one $\log$ in terms of others?

Can someone please help me with this logarithmic question? I know it’s easy, but I need to refresh my memory on how to do it. If $X=\log2$ and $Y=\log3$, express $\log0.6$ in terms of $X$ and $Y$ ...
2
votes
1answer
61 views

Choosing a branch for $\log$ when comparing $\prod_{n=1}^\infty(1+a_n)$ and $\sum_{n=1}^\infty \log{(1+a_n)}$

On Ahlfors on p. 191 he is talking about the relation between $\prod_{n=1}^\infty (1+a_n)$ and $\sum_{n=1}^\infty \log(1+a_n)$. He says: Since the $a_n$ are complex, we must agree on a definite ...
21
votes
1answer
359 views

A closed form for $\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx$

Is it possible to evaluate this integral in a closed form? $$\int_0^\infty\ln x\cdot\ln\left(1+\frac1{2\cosh x}\right)dx=\int_0^\infty\ln x\cdot\ln\left(1+\frac1{e^{-x}+e^x}\right)dx$$ I tried to ...
1
vote
2answers
3k views

What is the difference between logarithmic decay vs exponential decay?

I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.
2
votes
1answer
91 views

Which is greater: $n^{1.01} $ or $n\cdot log_{10}(n)$ ?

Can someone please explain how the right side can be less than the left side? I have plugged numerous numbers into n and every time I get the left side being less than the right side. My professor is ...
30
votes
1answer
867 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}\mathrm dx=\frac{\pi^2}8-\frac12$$
0
votes
1answer
752 views

Given the exponential equation $4^x = 64$, what is the logarithmic form of the equation in base $10$?

Given the exponential equation $4^x = 64$, what is the logarithmic form of the equation in base $10$? Would it be $\frac{\log_{10}64}{\log_{10}4}$?
2
votes
1answer
71 views

Solve: $\frac{1}{2}\log_{\frac{1}{2}}\left(x-1\right)>\log_{\frac{1}{2}}\left(1-\sqrt[3]{2-x}\right)$

Solve: $$\dfrac{1}{2}\log_{\frac{1}{2}}\left(x-1\right)>\log_{\frac{1}{2}}\left(1-\sqrt[3]{2-x}\right)$$ My try: Conditions identify: $\left\{ \begin{array}{l} ...
0
votes
1answer
255 views

Branch cut for $\log (iz)$ in the region $\{z:\mathrm{Im}(z)>0\}$

If someone could explain branch cuts and branch points to me that would be fantastic. I understand that a branch cut is a curve that we remove from the domain to make a function (usually a logarithm) ...
25
votes
3answers
1k views

Integral $\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}\mathrm dx$

I need your assistance with evaluating the integral $$\int_0^\infty\frac{1}{x\,\sqrt{2}+\sqrt{2\,x^2+1}}\cdot\frac{\log x}{\sqrt{x^2+1}}dx$$ I tried manual integration by parts, but it seemed to only ...
18
votes
2answers
876 views

Integral $\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$

Please help me to evaluate this integral: $$\large\int_0^{\pi/2}\frac{x}{\sin x}\log^2\left(\frac{1+\cos x-\sin x}{1+\cos x+\sin x}\right)dx$$
2
votes
3answers
96 views

Why isn't $2\log(-1)$ real?

In high school we learn that a $a\log[(x)] = \log (x^a)$ From this I would assume $2\log(-1) = \log [(-1)^2]$ However, the first is not real and the second is, according to my calculator and ...