Questions related to real and complex logarithms.

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0
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0answers
11 views

how to proceed next in this logarithmic inequality?

the question is $\frac{1}{log_4 (x+1)/(x+2)}$<$\frac{1}{log_4 (x+3)}$ I did the first step for defining the arguments of both sides and got $x\in(-3,-2)\cup (-1,\infty)$ next I did reciprocal ...
0
votes
2answers
30 views

finding out total digits in a large number

Is there any easy way to find out how many digits does the number $12^{400}$ have or such types of problems like how many digits the number $x^y$ have? ($x$ and $y$ are variables)
-1
votes
0answers
26 views

How do I obtain the running time for $T(n)=n^2 \sqrt{n}$?

I tried as, $$T(n)=n^2 \sqrt{n} =n^{\frac{5}{2}} $$ On expanding, $$ T(n)=n^{\frac{5}{2}}+n^{(\frac{5}{2})^2}+n^{(\frac{5}{2})^3}+\cdots +n^{(\frac{5}{2})^k} $$ Thus, for $T(1)$ $$n^{(\frac{5}{2})^k}=...
0
votes
2answers
18 views

How would I use the difference quotient on this logarithmic function?

This is no homework, it's for exam practice. Show that $\lim_{x\rightarrow 0}\frac{1}{x}ln(1+ax) = a$ where $a \in \mathbb{R}\setminus \left \{ 0 \right \}$ is chosen definitely / fixed (...
0
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2answers
66 views

How to differentiate $\ln(a^x)$?

Can someone give me the process to differentiate this (with respect to $x$)? $$ \ln(a^x) $$
0
votes
0answers
28 views

Why is the graph incomplete?

I've put this formula in google: log(2x-3) And it draws this graph: link to graph Why is the graph not being drawn all the way down? It is supposed to follow the ...
0
votes
1answer
19 views

p(a,c) vs p(a∧c)

In this paper: https://www.aclweb.org/anthology/J/J16/J16-2006.pdf, the author breaks the Pointwise Mutual Information of a phrase up into several components: They use the ...
2
votes
1answer
46 views

I stumbled on a relationship between ln(x) and estimated probability. Can someone help me locate or generate a proof?

Yesterday, I personally stumbled on the following relationship of ln(x): Say you have x number of checkboxes, and you randomly pick a position (p) between 1 and x. If the checkbox at position P is ...
5
votes
1answer
112 views

How do we prove that $4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8?$

How do we prove that $$4(3\sqrt2-4)=\prod_{n=1}^{\infty}\left({e^{2\pi(2n-1)}-1\over e^{2\pi(2n-1)}+1}\right)^8\tag1$$ Rewrite as, to keep it simple Let $a=e^{2\pi(2n-1)}$ $$4(3\sqrt2-4)=\...
1
vote
2answers
72 views

Calculate the value of $e$ from integral definition

Starting with the definition of $e$ as $$\int_1^e \frac{dx}{x} = 1,$$ how can I show that $e = 2.718\ldots$?
-1
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2answers
93 views

2 Weird questions

Seems like I'm full of weird mathematical questions! Last time I made a question about imaginary numbers. This time I have 2 seemingly unrelated questions. But nevertheless it's always good (and fun)...
0
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2answers
38 views

Prove $pq + 5(p - q) = 1$

Could you please explain me how to solve: If $p\:=\:\log _{12}\left(18\right)$ and $q\:=\:\log _{24}\left(54\right)$, $pq\:+\:5\left(p-q\right)\:=\:1$ I tried this way: $p = \frac{2\log\left(3\...
2
votes
2answers
60 views

Prove: $\log _{c+b}\left(a\right)+\log _{c-b}\left(a\right)=2\log _{c+b}\left(a\right)\cdot \log _{c-b}\left(a\right)$, where $a^2 + b^2 = c^2$

Could you help me proving this? $$\log _{c+b}\left(a\right)+\log _{c-b}\left(a\right)=2\log _{c+b}\left(a\right)\cdot \log _{c-b}\left(a\right)$$ where $c$ is the length of the hypotenuse of a ...
-1
votes
1answer
21 views

Geometric progression and logarithms

I would like to ask you for some help, solving that: 'The sum of three members of a geometric progression ($a, aq, aq^2$) is $62$ and the sum of their decimal logarithms $lg$ is equal to $3$. $a$ and ...
0
votes
0answers
49 views

Simplify $F(x) = \exp[-\ln^2x^h]$

I was wondering if the expression $F(x) = \exp[-\ln^2x^h]$ can be simplified even further? As you can see, the $\ln$ which is the natural logarithmic function is raised (and not its argument) to power ...
0
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0answers
22 views

Logarithm's inequality correctness

It is well known that for , the following holds: Now, given a set of n points, P, is the following term right for every and for every : If so, how can i prove that the term exists? And if it ...
1
vote
4answers
32 views

How do I solve this simultaneous equation that has the constant $e$ inside?

$28.8=24.5+Ce^{(-kt)}$ -(1) $28.0=24.5+Ce^{-k(t+ \frac{29}{60})}$ -(2) What I did so far: $24.5=28.8-Ce^{(-kt)}$ $24.5=28.0-Ce^{-k(t+ \frac{29}{60})}$ $28.8-Ce^{(-kt)}=28.0-Ce^{-k(t+ \frac{29}{60}...
0
votes
2answers
83 views

Logarithmic equation solution

If $ \frac {\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}$, then what would be the value of $a^{b+c}.b^{c+a}.c^{a+b}$? I'm unable to proceed.
4
votes
1answer
57 views

Proof of logarithm inequality without continuity

Showing that the logarithm function is continuous in its domain boiled down to proving $$\frac{x}{1+x}\le \ln(1+x)\le x \ \ \text{for all}\ x >-1.$$ There are quite a few proofs already online. ...
3
votes
1answer
31 views

How to solve the inequality $\log_2(4^x-2(2^x)+17)>5$?

Find the number of positive integers not satisfying the inequality $$\log_2(4^x-2(2^x)+17)>5$$ My approach: let $2^x=t$ then inequality is rewritten in form $$\log_2(t^2-2t+17)>5$$ then I ...
1
vote
1answer
18 views

Order of growth rate in increasing order

This question is related to maths, so I post here. Actually it's a computer science question and I am facing this type of question while learning Design and Analysis of Algorithms, but we all know ...
1
vote
1answer
69 views

Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
1
vote
1answer
47 views

Is this a pure imaginary number?

I've met this formula and I need to demonstrate that it is purely imaginary (it has no real part). $\frac{1}{2}\log(-\exp(i2\pi q))$, //for a real "input" q. As I don't know much about maths, what I'...
0
votes
2answers
93 views

If the integral of $c/x$ is $c.log(x)+C$ what is the base?

This question is a follow up to an answer I gave here: How to integrate $1/x$? After the algebra I said that 'This step of course gives the argument of $\log()$ the value $e$... and note that so far ...
6
votes
6answers
504 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log z)^...
0
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0answers
27 views

Combining a working hypothesis for odd perfect numbers with an inequality for logarithms

Euler's theorem for odd perfect numbers states that if there exists and odd perfect number, that is an odd positive integer $n$ satisfying $\sigma(n)=2n$, where $\sigma(m)=\sum_{d\mid m}d$ denotes the ...
3
votes
1answer
41 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
2
votes
0answers
36 views

Minimize a huge two-variable logarithmic-trigonometric-radical expression (MSU entrance early July 2016)

Minimize \begin{align}R(a,x)&=\sqrt{13+\log_a\left(\cos\left(\frac xa\right)\right)^2+\log_a\left(\cos\left(\frac xa\right)^4\right)}+\sqrt{97+\log_a\left(\sin\left(\frac xa\right)\right)^2-\...
-2
votes
2answers
76 views

What can be said about algebraic properties of this object? [closed]

The following holds: $$\frac1{\pi }\log \left(\frac{\tau +x}{\tau -x}\right)=\tan (\pi x)$$ for all $$-1/2< x < 1/2$$ What we can say about $\tau$?
0
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0answers
6 views

Find the relation between mean and variance for lognormal distribution giving as input mean and standard deviation of normal distribution

I am working with a Lognormal distribution with mean $m$ and variance $v$. I give as input $\mu$ and $\sigma$ of the relative normal distribution in order to calculate the cumulative Lognormal. Now, ...
9
votes
6answers
599 views

Has anyone talked themselves into understanding Euler's identity a bit?

What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
1
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3answers
2k views

Logarithm proof problem: $a^{\log_b c} = c^{\log_b a}$

I have been hit with a homework problem that I just have no idea how to approach. Any help from you all is very much appreciated. Here is the problem Prove the equation: $a^{\log_b c} = c^{\log_b a}$ ...
2
votes
4answers
69 views

Prove that $\int_0^1\frac{x^y-1}{\log x}\mathrm dx=\log(1+y)$

The title says it all - I currently can't find a good way to start. Tried rewriting it into a line integral, but I really don't see a way to solve this right now. I'd appreciate any hints.
0
votes
3answers
66 views

Nasty Limit with Logarithms

It is maybe a simple question but right now I am not able to see it. For $r,q,B>0$ and $x \in \mathbb{R}$, why is the following limit equal to $1$: $$\lim_{d\to 0^+}\exp\left[\left(\frac{d}{1-q}\...
0
votes
1answer
25 views

Lowest bound on logarthmic equation with floor

I have the following equation (log base 10): $$\frac{x}{10^{\lfloor \log x/10 \rfloor}}$$ how can I show what the maximum value of this expression can be? i.e. $\frac{x}{10^{\lfloor \log x/10 \rfloor}...
12
votes
6answers
10k views

log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$. So I was wondering where I have gone ...
1
vote
1answer
25 views

Finding value of $x$ in an A.P.

If $1$ , $\log_{9}(3^{x+1} + 2)$, $\log_{3}(4⋅3^{x}-1)$ are in A.P. , then $x$ equals ?
0
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1answer
39 views

Rearranging log expression containing division and subtraction

I have $$ \log\left(\frac{b\exp(a)}{1 - b\exp(a)}\right)$$ and I try to find the shortest representation. I found $$ 0 - \log(\frac{1}{b\exp(a)} - 1)$$ and $$ a - \log(b - \exp(a))$$ I'm ...
29
votes
5answers
6k views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$ [closed]

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
3
votes
2answers
31 views

logarithm transition for the population growth equation

Analyzing the growth population equation I came across with the below transition $$\frac{d \ lnN}{dt}=\frac{dN}{N dt}$$ which I don't quite follow. Can anybody clarify this please?
1
vote
1answer
110 views

How to evaluate $\log(1 - x)$ in terms of $\log(x)$?

I can do this using the following relation: $$\log(1 - x) = \log(1 - \exp(y))$$ Here $y = \log(x)$ is always a negative number. However, I was wondering whether it's possible to compute $\log(1 - x)$...
0
votes
1answer
41 views

Solving Equation With Logarithm Argument Being a Variable

$$8n^2 = 64n\log_{2}n$$ Been a while since I have used logarithms. I am actually comparing the running time of two algorithms and can obviously solve graphically but for the life of me can't remember ...
0
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0answers
25 views

Exponential equation involving different bases.

Alright, I know you can get the solution $ x=2 $ just by giving the equation a glance, but I am asking for a rigorous proof here. All mathematical tools are encouraged (no syllabus limitation). $$ 5^{\...
0
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1answer
37 views

Holomorphic branch of square root of $f$

Let $f(x) = (z-\frac{1}{z})$ , $z \in \mathbb{C}$\ {${0}$} Let $F$ be a holomorphic branch of the square root of $f$ that is defined at $z=2$ and has the value $\sqrt{3/2}$ there. GIve an explicit ...
0
votes
2answers
2k views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
0
votes
1answer
20 views

Creating a logarithmic function with known $x-y$ axis intersections

I know that to plot a straight line that intersects the axis in point $(x_1,y_1)$,(x_2,y_2)$ one can use this equation. $$(x_2-x_1)\cdot (y-y_1)=(y_2-y_1)·(x-x_1)$$ To be more specific if i want a ...
2
votes
2answers
37 views

How to solve the equation $\log_{2x+3}(6x^2+23x+21)=4-\log_{3x+7}(4x^2+12x+9)$?

How to solve the equation $\log_{2x+3}(6x^2+23x+21)=4-\log_{3x+7}(4x^2+12x+9)$ ? Can someone please tell me a few steps as to how to approach these category of problems? I know $2x+3>0$ and $3x+7&...
0
votes
1answer
31 views

non-standard exponential-squared fog attenuation [closed]

I inherited a formula that I'm hoping to simplify. $d = \frac{\sqrt{-\log_2(t)}}{f\sqrt{\ln(2)}}$ Any ideas? Thanks, Jason EDIT (for context): This formula determines the exponent for exponential-...
0
votes
0answers
52 views

log of integral?

Let's consider a mid-point rectangular integration of a function $f(t)$ and suppose we are interested in: $$\log\left(1 - \int_0^t f(z) \, dz\right) \approx \log \left(1 - \sum AUC_{\text{rect}_i} \...
1
vote
5answers
250 views

Is the natural logarithm actually unique as a multiplier?

The Wikipedia page on the natural logarithm says: 'Logarithms can be defined to any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from ...