Questions related to real and complex logarithms.

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Finding the equality of the natural logarithm to the limit and the infinite series (proof)

I'm trying to proof this equality which I found on this website: Euler-Mascheroni constant expression, further simplification $$\ln(n)=\lim_{M\rightarrow\infty}\sum\limits _{k=1}^{M}\sum\limits ...
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3answers
94 views

Are there any series that converge to $log_{2}$?

I am wondering, are there any sums that converge to a $log_{2}$ of something? I know this question is vague, but I would love to see some examples since I am studying worst case complexity in ...
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0answers
48 views

Give me the proof of this equality!? [duplicate]

I would like that someone can tell me stap for stap (mathematically proof) that this equality is true! I hope someone can help me and if you can I would be very thankfull ;) ...
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4answers
40 views

Why log(A) can be seen as %ΔA in economics?

We usually see deduce in economitrics changes $lnY$ to $\%\Delta Y$, say we have Cobb-Douglas function like this $Y=AK^αL^{1−α}$, in one of the book I read, it is changed to : But why can we do like ...
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1answer
12 views

Stuck on Double-Variabled Logarithm when solving this Sequences and Series question

I'm able to get to an inequality for the sum of the arithmetic sequence greater than the sum of the geometric sequence, and have solved the inequality by guess and check and have verified the ...
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1answer
67 views

A Logarithm Integral II [on hold]

Does the integral \begin{align} \int_{0}^{1} (1-t)^{2} \, \ln^{k}(1-t) \, \ln^{m}(t) \, dt \end{align} have a compact form for $m = 1$, and $m=2$ ?
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0answers
97 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification ...
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1answer
8 views

Find all solutions for a complex logarithm

$\log z = 6i$ I am working on a problem very similar. What I am seeing $\log z = \ln|z| + i(\theta + 2\pi n)$ for $n\in\mathbb{Z}$ What I am curious about, as if seen obvious to me that $ \log ...
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0answers
82 views

How can we proof that this is equal? About $ln(n)$

I found this on this website (Euler-Mascheroni constant expression, further simplification) without any explaining why this is equal can someone give me that? ...
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2answers
48 views

Is $g(x)=\log x$ convex function?

The graph of convex function is : In a book it is written that $g(x)=\log x$ is strictly convex function. So i searched for graph of $g(x)=\log x$ and found that Though it has been said that ...
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0answers
36 views

Analytically solving complicated integral involving logarithms.

I already asked a similar question a week ago and the comment I got helped me a lot with my progress, so that I now have new question to ask. I am stuck with solving a complicated integral and would ...
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2answers
42 views

Different results when integrating $1/(x \ln(x))$ partially/by substitution.

By substitution I get $ln(ln(x))$. Partially something completely different: $$\int \frac{1}{x \ln(x)} = \int \frac{1}{x} \frac{1}{\ln(x)} dx=\frac{\ln(x)}{\ln(x)} - \int -\frac{1}{x \ln(x) ^2} dx$$ ...
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1answer
44 views

A Trig Integral

Does the integral \begin{align} \int_{0}^{\pi/2} \cos(x) \, \ln\left( \frac{1 + a^{2} \sin(x)}{1 - a^{2} \sin(x)} \right) \, dx \end{align} have a closed form and what is changed if the limits are ...
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2answers
23 views

Prove the logarithmic inequality

Prove that: $(\log_{24}{48})^2+(\log_{12}{54})^2>4$ I tried to put $t=\log_23$ and get the equation $6t^4+32t^3+22t^2-84t-74>0$. But I can't do anything with it...
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1answer
42 views

Inequality with Logarithms!

I need some help solving this inequality for a question involving the number of bounces, $n$, of ball such that the max. height of the ball is less than 5cm. This is the equation I have gathered from ...
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0answers
36 views

What expression would one use to express the following? [on hold]

130 - 120 = -1.39 120 - 110 = -1.51 110 - 100 = -1.65 100 - 90 = -1.82 90 - 80 = -2.04 80 - 70 = -2.31 70 - 60 = ? What expression expresses the above? I think it's exponential.
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1answer
50 views

A definite integral contianing ln(x)

everyone, I met a tough definite integral as follows, $$I = \int\limits_1^\infty {\frac{{\ln x}}{{{{\left( {x + a} \right)}^m}{{\left( {x + b} \right)}^{n + 1}}}}} dx,$$ where $a$ and $b$ are ...
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1answer
34 views

Complex logarithm function

I need some help understanding the logarithm function in complex plane. Let $w,z\in \mathbb{C}$. Define $$w=e^z$$ when $$z=\log(w).$$ Now I understand the representation of ...
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1answer
67 views

Simplifying a log of a log

I have a summation series that unfortunately involves a log of a log. It looks like the following (assume all $\log$ are log base $2$): $$ \sum_{i=1}^k \log\log\frac{n}{2^{k-i}} $$ I'd like to ...
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1answer
39 views

Solving Log equation using master theorem

I`m studying Master Theorem, and I got stuck in the case 3. The example is : T(n) = 3T(n/4) + nlogn. I have no idea how my teacher got the final value, c = 3/4, based on the equation below : 3*[n/4 ...
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2answers
45 views

$\sum x^n/n$ — why does it equal $\log(\frac {1}{1-x})$?

Define the function $D(x) = x + x^2/2 + x^3/3 + \cdots$ I found out during a brief exchange with a friend that this sum equals $\log\left(\frac 1{1-x}\right)$ for $|x| < 1$. He had learned it in a ...
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2answers
45 views

An equation not so easy! Help me with this logarithmic equation!

I need help with the following problem analysis, if someone could I resolve all steps. Let $a>0$. Determinate the number of solutions of the equation $ax^2 = \log x$, according to the values the ...
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5answers
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How can I solve $\log_2(x+1)-\log_2(3x+1)=2$ for $x$? [closed]

Can someone help me here with this log equation? $$\log_2(x+1)-\log_2(3x+1)=2$$
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2answers
32 views

Trying to solve this question I'm new to logs [closed]

How to solve 3log(a+b/2)=log a+2 log b
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1answer
14 views

Solving exponential equation - Order of operations

Hopefully this should be a quick questions. When solving the exponential equation 5 * 2^(u/2) + 30 = 600 Why do you subtract 30 first and not divide 600 by 5? The order of operations indicates that ...
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2answers
25 views

Verify that Log$(z^{w}) = w$Log$z$ + $2\pi i n$

The symbol "Log" denotes the complex logarithm. Let $w$ be a complex number so that $w = u+iv$ for some reals $u, v.$ We have $$\mbox{Log}(z^{w}) = \log |z^{w}| + i\arg (z^{w}) = u\log |z| - v\arg ...
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2answers
246 views

For what $n$ does $[\log_21]+[\log_22]+[\log_23]+\dotsb+[\log_2n] = 1538$? [duplicate]

I just can't solve this problem in spite of doing a whole book on logs and inequalities Where $[\dotsc]$ denotes the greatest integer function, what is the value of the natural number $n$ ...
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2answers
44 views

Tricky Logarithmic Inequality Problem

I am having a problem solving this question - If $\log_{\frac{1}{\sqrt{2}} }{\sin{x}}>0$, $x\in [0,4\pi]$,then number of values for chating which are integral multiples of $\pi/4$,is A-6 B-12 ...
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3answers
64 views

Integral of $\log(\sin(x)) \tan(x)$

I would like to see a direct proof of the integral $$\int_0^{\pi/2} \log(\sin(x)) \tan(x) \, \mathrm{d}x = -\frac{\pi^2}{24}.$$ I arrived at this integral while trying different ways to evaluate ...
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0answers
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Taking integral of the complex logarithm using fundamental theorem?

Is it valid to do this? I have $f(z)= z^i$,and $F(z)=\frac{z^{i+1}}{i+1}$ and assuming we're using principle values of $f$ and $F$ would it be correct to say that: $\int_{-1}^{1} f(z) dz = ...
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0answers
25 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
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1answer
40 views

Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x $cannot be ...
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0answers
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Integration of Exponential and Logarithms, $\int_{z-1}^z \log(\frac{1}{z-y}) \exp (-| y| ^{3}) \, dy$

The integral I am dealing with is: $$\frac{3}{2 \Gamma \left(\frac{1}{3}\right)}\int_{z-1}^z \log \left(\frac{1}{z-y}\right) \exp \left(-\left| y\right| ^{3}\right) \, dy$$ where $z\in \mathbb{R}$ ...
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2answers
39 views

Determine the convergence or divergence of $\sum_{2}^{\infty}\frac{1}{(\log n)^{s}}$, where $s \in \mathbb{R}$ is given.

Since $$\frac{1}{(\log n)^{s}} > \frac{1}{n^{s}}$$ for large $n$, if $s \leq 1$ then $\sum_{2}^{\infty}\frac{1}{(\log n)^{s}}$ diverges. But for $s > 1$ I have not yet figured out a proof.
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2answers
115 views

Can anyone solve this equation?

Having trouble working this one out: $$25^x + (2 .5)^x = 35$$ Any help would be appreciated.
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0answers
24 views

Taking the logarithm of a periodic function

I've been wondering how we take the logarithm of a periodic function. At least I think that's what I've been wondering - but I may have confused the terminology. Anyway, take, for example, the ...
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1answer
33 views

Minimum value of a Logarithmic equation

What is the minimum value of $$\log_a(x)+ \log_x(x) $$ where $0\leq a\leq x.$ I do not understand why my book says the answer is $2$ because when i take $a=0.1$ say and $x =0.2$ I get $\approx ...
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2answers
51 views

Is it true that $\int_{0}^{1}(1+x^{2})^{-1/2} = \log (1 + \sqrt{2})$?

Since $$D^{-1} (1 + x^{2})^{-1/2} = \sinh ^{-1} (x) + C,$$ is it true that $$\sinh ^{-1} x + C \big|_{0}^{1} = \log (1 + \sqrt{2})?$$ What relates $\sinh^{-1}(\cdot )$ to $\log(\cdot )$? Here ...
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2answers
26 views

Help to find the best lower bound function for a given set of data, based in the natural logarithm function

I am trying to find a lower bound function for a set of data I have, and I am struggling with it. In the following graph the blue color is the set of data and the red color is my lower bound function. ...
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1answer
15 views

The position of significant digits and Logarithms relationship…

I am unable to solve the following question has i don't understand what the relationship is between significant figures and Logarithms. Q-If $\log_{10}(7)= 0.8451$ then the position of the first ...
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1answer
28 views

Stuck with understanding transformation step in calculating limit of $n(\sqrt[n]{a}-1)$

Although this question has already been asked in general ( $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$) , my question is different, because I am stuck with a specific transformation step: ...
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2answers
33 views

Integration of logarithm

$\int \ln(\ln \sqrt{x})^{\ln (x)}dx$ how should I integrate this? I think it can't be integrated. I don't know.
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1answer
47 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
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2answers
50 views

What is this equation?

I ran across this equation for use in web code here and am desperately wanting to know if any portion of it or the whole thing is a standard equation somewhere. This is the best I could do ...
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2answers
30 views

rewrite logarithmic expression

I have this logarithmic expression 2 logb 6 + (1/2) logb 25 - logb 30 and have to rewrite it as logb of one number. I just don't understand how to do this. help please.
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1answer
32 views

How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$. Or to put it in a more formal way: For every $ε>0$, there exists a constant $N$ for ...
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1answer
32 views

Simplification of a logarithm expression

I need to verify the answer of a logarithm expression (note, I'm not a student). I managed to get through high school and college without ever having a math course that taught logarithms--I don't ...
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1answer
20 views

Deriving a function with logarithmic terms

Let $L(X) = \exp(\sqrt{\log X \log \log X})$ Prove that if $c > 0$,$ Y = L(X)^c$, and $u = \log X/ \log Y$ , then $$u^u = L(X)^{(1/2c)(1+o(1))}$$ I've tried to write $u^u = (\log X/ \log ...
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1answer
28 views

Logarithm, Just need help understanding what this question is asking. Not looking for an answer.

In my foundations of computing class, we were given a logarithm question which i don't quite understand. This is the question. Given the logarithmic table values of the numbers x and y are ax and ay ...
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2answers
50 views

Deriving properties of the logarithm from its integral representation

Suppose we define: $$\ln(x) = \int_{a}^{x} \left[ \frac{1}{r} dr\right]$$ Such that $$ \ln(1) = 0, \ln(e) = 1$$ How does one derive all the properties of the logarithm from the properties of the ...