Questions related to real and complex logarithms.

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0
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2answers
28 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 ...
2
votes
0answers
119 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 ...
1
vote
1answer
12 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 ...
0
votes
0answers
86 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? ...
3
votes
2answers
484 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 ...
0
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2answers
48 views

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

By substitution I get $ln(ln(x))$. Partially something completely different: $$\int \frac{1}{x \ln(x)} = \int \frac{1}{x} \frac{1}{\ln(x)} dx=\frac{\ln(x)}{\ln(x)} - \int -\frac{1}{x \ln(x) ^2} dx$$ ...
0
votes
1answer
58 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 ...
1
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2answers
30 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...
1
vote
1answer
53 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 ...
2
votes
1answer
55 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 ...
3
votes
3answers
153 views

How to solve:$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...
0
votes
1answer
44 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 ...
1
vote
1answer
72 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 ...
1
vote
1answer
43 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 ...
1
vote
2answers
50 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 ...
0
votes
1answer
21 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 ...
0
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2answers
31 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 ...
-4
votes
2answers
266 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 $n$ ...
0
votes
2answers
55 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 ...
2
votes
3answers
82 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 ...
1
vote
0answers
26 views

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 = ...
2
votes
0answers
27 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 = ...
0
votes
1answer
44 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 ...
4
votes
1answer
51 views

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 $z\in \mathbb{R}$ ...
1
vote
2answers
40 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.
0
votes
0answers
36 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 ...
0
votes
1answer
38 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 ...
0
votes
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 ...
0
votes
2answers
41 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. ...
1
vote
1answer
25 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 ...
2
votes
1answer
29 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: ...
-1
votes
2answers
36 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.
1
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1answer
62 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 ...
0
votes
2answers
64 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 ...
0
votes
2answers
41 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.
1
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1answer
33 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 ...
0
votes
1answer
35 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 ...
0
votes
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 ...
1
vote
1answer
44 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 ...
1
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2answers
51 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 ...
1
vote
1answer
29 views

proving with a sequence

The question is : Show that if $n$ is a power of $2$, then $$\sum_{i=0}^{\log_2n-1}2^i=n-1\;.$$ Tried induction at first and tried to prove it on 2n but nothing came out of it. Then i tried ...
0
votes
3answers
30 views

if $x>1$ and $\log_2x,\log_3x,\log_x16$ are in G.P then what is x $=$

if $x>1$ and $\log_2x,\log_3x,\log_x16$ are in Geometric progression then what is x equal to? Solution: $(\log_3x)^2=\log_2x\times\log_x16=\log_216=\log_22^4=4$ $\log_3x=2 or x=3^2=9$ so my doubt ...
-3
votes
1answer
89 views

Check whether a function is one-to-one and onto

If $f(x) = \log_{x^3}\left(\sqrt{x}\right)$, check whether $f$ is one-to-one and onto where $x\in R^+\setminus\{1\}$. Also write the range of $f$. Alright, if $f(m) = f(n)$ and if we would prove m=n ...
0
votes
6answers
64 views

Limit of log functions

I need help solving this problem. $$\displaystyle \lim _{x\to 0}\frac{\log\left(1+7x\right)}{5x}$$
1
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3answers
45 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
1
vote
3answers
41 views

How do you solve this logarithmic equation?

While reading through my textbook, I came across this particular equation: $$ x = x\log (y) + \log (y) $$ But they solve it by doing this: $$ x = x\log (y) + \log (y) $$ $$ x = (x + 1)\log(y) $$ $$ ...
0
votes
3answers
22 views

Derivate a logaritmic function

Let's take $ f = \ln(x) $. The derivate is $ f' = 1/x$. However $g = \ln(50x) $ has the same derivate $f' = g'$. How come? If I where going to derivate $g$ I would substitute $x$ for $t$: $g = ...
0
votes
2answers
53 views

Is $\ln n$ transcendental for all rational $n>1$?

I know that $\ln n$ is transcendental for all integer $n>1$. But does this still hold for non-integer rational values of $n>1$? For example, is $\ln 1.5$ transcendental? EDIT: Somehow managed ...
0
votes
1answer
36 views

logarithmic differentiation issue

Trying to understand a solution I was given to a problem I was told to use logarithmic differentiation on. $$ 1/x(x+1)(x+2) $$ and I know that $$log((ab)/c) = log(a) + log(b) - log(c)$$ So I tried to ...
0
votes
1answer
45 views

Implications of redefining base natural logarithm constant e

Disclaimer: I'm no math expert! I understand that the constant $$e$$ is expressed as follows: $$e = \sum_{n=0}^{\infty} \frac1{n!} = 1 + \frac1{1*1} + \frac1{1*2} + ...$$ What would be the ...