Questions related to real and complex logarithms.

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3
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1answer
77 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
0
votes
1answer
40 views

trouble with infinite values from exp() and log()

I'm writing a function for Gaussian mixture models with spherical covariance structures--ie $\Sigma_k = \sigma_k^2 I$. This particular function is similar to the ...
2
votes
0answers
78 views

What did “logarithm” initially mean? [duplicate]

I just read that logarithms were not initially defined in terms of their inverse relationship to exponential functions (and that Euler was the first to develop them in this way). So how were they ...
0
votes
1answer
70 views

How to simplify $3^{(2\log_335)}$

$3^{2\log_35}$ How do I simplify this? This is what I have done so far: $2\log_35=\log_35^2=\log_3(25)$ $3^{\log_3(25)}$ What do I do from here? And the answer is one of these mixed solutions: ...
3
votes
2answers
76 views

Non-integral power of a singular matrix

I know, that if $A$ is nonsingular matrix, so $\det{A} \ne 0$, then $A^p=\exp\left(p\ln A\right)$ is true for any real exponent, but what about if $A$ is singular? Then $A$ has a zero eigenvalue, so ...
1
vote
1answer
40 views

Logarithmic Contest Question

The Problem was as follows: Define $\log*(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to $1$. For example ...
6
votes
3answers
141 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
0
votes
0answers
39 views

Are generating functions ever analytic for logarithmic series?

Given a series $s_n = \ln(n) f(n)$ where $f(\cdot)$ is an elementary analytic function which does not involve the logarithm. More precisely $f$ can have simple poles but no branch cuts or essential ...
0
votes
1answer
30 views

Exponential percentage decrease based on time

I have a bar that shows the time left for a task to finish and I want it to decrease faster as it gets closer to the end time. Example: Let's assume that the total time required for Task A to ...
2
votes
1answer
70 views

$\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ check my answer!

I would like someone to review my solution please, the original question is to calculate $\iiint \frac{1}{x^2+y^2+(z-2)^2}dA$ where $A=\{x^2+y^2+z^2 \leq 1\}$ What I did: First I changed variables ...
2
votes
0answers
47 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
0
votes
2answers
35 views

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in ...
0
votes
1answer
33 views

Convergence $\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$

Help please. I need to check the convergence $$\sum_{n=2}^{\infty} { \frac{\sqrt[n]{n^{p}}}{n\ln{n}} }$$Tried with Leibniz, but can't check monotony.
2
votes
3answers
98 views

Summation with Ceilinged Logarithmic Function

According to Johann Blieberger's paper - "Discrete Loops and Worst Case Performance" (1994): $$ \sum_{i = 1}^{n}\left \lceil \log_2{(i)} \right \rceil = n\left \lceil \log_2{(n)} \right \rceil - ...
0
votes
1answer
30 views

Asymptotic behaviour of a couple of special functions (features exponentials and logarithms)

I'm dealing with a couple of functions: $n \log n$, $( \log \log n)^{ \log n}$, $( \log n)^{ \log \log n}$, $n e^{\sqrt{n}}$, $( \log n)^{ \log n}$, $n 2^{ \log \log n}$, $n^{1+1/( \log \log ...
0
votes
1answer
68 views

How to simplify $\ln^2\left(x\right)+2 \ln x-3$

I dont know how to simplify $\ln^2\left(x\right)+2 \ln x-3$ I dont know how to get $(\ln(x)+1)(\ln(x)+3)$ But I am stuck and don't really know how to do that. I tried something like this: $2\ln ...
3
votes
4answers
75 views

Solve for $x$ in $2\log(x+11)=(\frac{1}{2})^x$

Solve for $x$. $$2\log(x+11)=(1/2)^x$$ My attempt: $$\log(x+11)=\dfrac{1}{(2^x)(2)}$$ $$10^{1/(2^x)(2)}= x+11$$ $$x=10^{1/(2^x)(2)}-11$$ I'm not sure what to do next, because i have one $x$ in ...
1
vote
1answer
76 views

Upperbound for $ \sum _{i=1}^N a_i\ln a_i $

It's easy to prove that following upperbound is true: $\sum_{i=1}^N a_i\ln a_i \le A \ln A$, where $\sum_{i=1}^N a_i=A$ and $ a_i\ge 1$ I'm wondering, is there stronger upperbound?
0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
0
votes
1answer
17 views

Log-space probability of a log-space probability not occurring

Normally the probability of some probability $p$ not occurring would be $1-p$. However, I'm working with very small probabilities and therefore must work with $p$ in $\log$ space (Ie. I'm working with ...
3
votes
1answer
46 views

Could somebody validate my proof regarding the limit of $\ln(x_n)$ when, $x_n$ tends to $a$?

So, let me cearly state the problem: Let $(x_n)$ be a convergent sequence, with: $ x_n > 0 $, $\forall n$, n natural number, and $x_n \to a$, with $a>0$. Then $\ln{x_n} \to \ln{a}$. Here is my ...
1
vote
1answer
36 views

Solving $Ae^x=Bx$ analytically, where $A$ and $B$ are constants?

This equation mixes both exponential terms and linear terms, something which I do not know how to deal with. Any pointers?
0
votes
1answer
16 views

Struggling with a Form of Logarithm question during my revision

I am doing AS Mathematics In the UK under the examining board edexcel. I came across this question in a List of exam questions given to me by my teacher However I can't work out how to do it. ...
2
votes
1answer
57 views

How did Napier rounded his logarithms?

How did Napier round his logarithms? Wikipedia says: By repeated subtractions Napier calculated $(1 − 10^{−7})^L$ for $L$ ranging from 1 to 100. The result for $L=100$ is approximately $0.99999 = ...
1
vote
2answers
189 views

Properties of Logarithms

How do you simplify the following expression? $$\log\left(3^{(5^7)}\right)$$ I know that logarithms are like the inverse of exponents, but are there any tricks to simplify powers inside logarithms? ...
1
vote
2answers
50 views

Finding unknown in log equation

I was given a log equation: $$D = 10 \log (I/I_0) $$ $I$ is the unknown in this case, $I_0 = 10^{-12}$ and $D = 89.3$. I did the following steps: $$ \begin{aligned} \ 89.3 &= 10 \log ...
6
votes
4answers
164 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
0
votes
1answer
35 views

How to rewrite this logarithmic update rule

I tried to rewrite the equation given below. I get stuck getting rid of the $ P(n|z_{1:t})$ on the left side. How can this be done? $$ P(n|z_{1:t}) = \left[1+ \frac{1-P(n|z_{t})}{P(n|z_t)} ...
1
vote
1answer
27 views

Equal spaced points in a logarithmic graph

I am plotting a graph with the x-axis as logarithmic. I want to select 10 point that are equally spaced in a logarithm scale. How can I determine the values if we have the range from 100 to 10000?
0
votes
2answers
29 views

Logarithmic Properties for Equation [closed]

What are the steps/logarithmic properties used to solve the equation 2^n=n^8? Thanks!
0
votes
1answer
37 views

How is $\arctan(\sinh(x * \pi))$ the inverse of $\log(\tan(x)) / \pi$

How is $$\arctan(\sinh(x * \pi))$$ the inverse of $$\frac{\log(\tan(x))}{\pi}$$ What is the relationship between $\log(x)$ and $\sinh(x)$. I guess is what my real question is.
1
vote
3answers
55 views

Why $\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$

Why $$\ln 2=\ln 1.075^t\implies \ln 2=t\ln 1.075$$
4
votes
2answers
57 views

Solve numerical system of nonlinear equations?

I need to solve a nonlinear system of equations that looks like this ...
-2
votes
1answer
47 views

Prove that $\log n = O(\log^2 n)$

Trying to solve this, but I can't seem to figure it out. Its fairly straight forward.
2
votes
1answer
29 views

Exponential continuous growth $\ln a$ vs. $r$? Huh?

So, given a simple population continuous growth problem, it seems that the entirety of the internet uses $P=P_0e^{rt}$ where $P$ is the population over time, $P_0$ is the initial population, $r$ is ...
0
votes
0answers
12 views

Find the value of x if $\log_{2}({5 * 2^x + 1})$, $\log_{2}({2^{1 -x} + 1})$ and 1 are in arithmetic progression

My try: $1 - \log_{2}({2^{1 -x} + 1})$ = $ \log_{2}({2^{1 -x} + 1}) - \log_{2}({5 * 2^x + 1})$ $10 * 2^x + 2 = (2^{1 -x} + 1 )^2$ Let $2^x = t $ $10t^3 + t^2 -4t -4 = 0$ But I ...
1
vote
2answers
223 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
1
vote
2answers
24 views

no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$

The given equation is $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$ I took $\log_{2}{x}$ = $t$ and then rewrote the given equation as $$x^{3t^2 + 4t - 5} = \sqrt{2}$$ ...
0
votes
1answer
28 views

Number raised to log expression

I am struggling with what I think should be some a basic log problem: Show that $3^{log_2n} = n^{log_23}$ I know that $3^{log_3n} = n$ and $log_2n = {log_3n}/{log_32}$ I was attempting something ...
1
vote
0answers
10 views

How to find the highest possible value and time to achieve it with a set increment and percentage decrease plus rounding?

I'm trying to answer a preparation question in math, but I don't get it, so an formula with an explanation would be very helpful! In a simple system measuring recent user activity each user have one ...
0
votes
1answer
42 views

If $\log_{30}{3} = c$ and $\log_{30}{5} = d$ then the value of $\log_{30}{8} $ is??

I attempted the following: $\log_{30}{8} = 3\log_{30}{2}$ $\log_{30}{3} = c$ is equivalent to $3 = 30^c$ $\log_{30}{5} = d$ is equivalent to $5 = 30^d$ What should I do further?? Is ...
1
vote
1answer
54 views

Why does this equation work?

let $ P(x) := \sum_{p \leq x} Log [p]$, then we have $P(2^{k+1}) = \sum_{i=0}^k ( P(2^{i+1}) - P(2^i)) < 2 \cdot Log[2] \cdot (1 + 2 + 4 +... + 2^k) \leq 4 \cdot Log[2] \cdot 2^k$. Why does ...
1
vote
1answer
97 views

Anti-log of a number

If we accept both positive and negative values for the square root of a number, then can the anti-log of a number be negative?
1
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2answers
34 views

Proof by induction using logarithms

I have come across a question while studing for my exams prove $$\log_2 x < x \text{ when }x>0$$ I know I have to solve it using a base case eg when $x=1$ then assume a inductive step $x=k$ is ...
2
votes
2answers
50 views

Expressing $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$

Express $\ln \sqrt[3]{54}$ in terms of $\ln 2$ and/or $\ln 3$ I know that $\sqrt[3]{54}=54^{1/3}$ but otherwise I don't know how to address these types of problems. How do I solve this, and is ...
0
votes
1answer
45 views

Distribution of log-log linear regression

Edit: Sorry yeah not too clear, probably posted this too late at night... Essentially I have data which appears to be in exponential form - a log-log graph put it close to a straight line. Using R, I ...
0
votes
1answer
27 views

Upper bound of natural logarithm

I was playing looking for a good upper bound of natural logarithm and I found that $$\ln x \le x^{1/e}$$ apparently works: Can someone give me a formal proof of this inequality?
1
vote
1answer
50 views

Rewriting in $y=A_0\cdot e^{at}$

How do you rewrite $y = −8(1.589)^{t − 3}$ in $y=A_0e^{at}$ form for appropriate constants $A_0$ and $a?$ For other problems I took the $\ln$ of the number inside the parenthesis. So for example I ...
1
vote
1answer
26 views

Complex Logarithm Derivation

I don't understand how the definition of the complex logarithm was derived. It is $ log(z) = ln|z| + i Arg (z) $, where $ z = x + iy $. I've tried all sorts of method to find this definition but ...
2
votes
2answers
31 views

Obtain the set of real numbers $c$

Show that there exists a positive real number $x \ne 2$ such that $\log_2 x ={x\over2}$ . Hence obtain the set of real numbers $c$ such that $\log_2 x\over x $$= c$ has only one real solution.