Questions related to real and complex logarithms.

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5
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2answers
240 views

Identities derived from Euler's Identity

I am just learning about Euler's identity. When messing around with it, I am getting some unsettling identities, which, I believe, is probably due to my lack of application of certain rules. Starting ...
1
vote
3answers
41 views

Finding intersection between functions containing logarithms

At what point does $8x^2$ and $64x\log(x)$ intersect? I'm trying to catch up on my math, but this stumped me. Something awry in my understanding of logarithms. I figured that I could equate both ...
1
vote
2answers
29 views

If a>0 a is not equal to 1 then the equation

$2 \log_x (a)+ \log_{ax} (a) + 3\log_{a2x} (a)=0$ then he equation has how many real roots?? The above problem is a quadratic equation problem.
1
vote
0answers
22 views

Sum of Logarithms of expoenetial functions

Given a matrix $P \in \Re^{n \times d}$, and a column vector $\theta \in \Re^d$. Assume that $\sum\limits_{i=1}^n \ln{(1+e^{P_i\theta})} \leq 1$, where $P_i$ is the $i^{th}$ row in $P$. What can be ...
0
votes
1answer
28 views

Difference between branches $[-\pi, \pi)$ and $[\pi, 3\pi)$ of the complex logarithm

I think that both branches just exclude the negative part of the real line. So what's the difference between them then?
0
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0answers
25 views

norma distribution and log-normal distribution

I often see when people analyzing data, they assume data has either normal or log-normal distribution, and trying to fit data into a distribution for the convenience of data analysis (e.g. by ...
-2
votes
1answer
28 views

How do you calculate what $k$ is? [on hold]

How do you calculate what $k$ is? $$\frac{d-1}{k} = \log \left(\frac{21}{100} d\right)$$ And how do I get the integer value?
0
votes
2answers
48 views

Natural logarithm question 1

i tried to derive logistic population model, and need to integrate this $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t$. here is my solution $\int \frac{\frac{1}{k}}{1-\frac{N_t}{k}} dN_t=\int \frac{...
1
vote
4answers
89 views

Show that $\log(1+y) \approx y- \frac {y^2}2 + \cdots$ without Taylor Series

For small $y$, prove that $\log(1+y)\approx y -\frac {y^2}2 + \cdots $ I have no idea to solve it.
1
vote
3answers
37 views

Calculating integer solutions of a logarithmic equation

The question asks: Calculate the integer solutions of the equation $\log_2(x+2)+\frac12\log_2(x-5)^2=3$ To me, this is trivial if solved in the following way: $(x+2)+(x-5)=2^3$ $2x-3=8$ Answer: $x=\...
0
votes
1answer
37 views

how to proceed next in this logarithmic inequality?

The question is $$\frac{1}{\log_4{\left(\frac{x+1}{x+2}\right)}}<\frac{1}{\log_4{(x+3)}}$$ I did the first step for defining the arguments of both sides and got $x\in(-3,-2)\cup (-1,\infty)$ ...
-2
votes
1answer
63 views

How to solve this hard equation? [on hold]

I have read this equation on a journal: $3\left(\log _3( {\sqrt {2 + x} + \sqrt {2 - x} }) \right)^2 + 2{\log _{\frac{1}{3}}}\left( {\sqrt {2 + x} + \sqrt {2 - x} } \right)\cdot {\log _3}\left( {9{...
0
votes
2answers
31 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
27 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
21 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
votes
0answers
32 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
20 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 ...
1
vote
2answers
73 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 $e = 2.718\ldots$?
0
votes
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
50 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
votes
0answers
23 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}...
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
32 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
19 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
75 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
votes
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)...
1
vote
1answer
48 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
0answers
28 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 ...
2
votes
0answers
37 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-\...
0
votes
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, ...
2
votes
4answers
71 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}\...
5
votes
1answer
114 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)=\...
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}...
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
votes
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 ...
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 $\log(1 - x)$...
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?
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
votes
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
votes
1answer
21 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
38 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
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
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} \...
0
votes
2answers
28 views

Proof of equation $\ln y=-kt+c$ is same as $y=c*e^{-kt}$

I'm trying to prove that one is the same as the other : $$\ln y = -kt+c$$ $$y=ce^{-kt}$$ Where c is undefined and k is defined constant. I got as far as: $$y=e^{-kt+c}$$ So by what rule would c be ...