Questions related to real and complex logarithms.

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5
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2answers
454 views

continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
5
votes
3answers
91 views

What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
5
votes
2answers
408 views

Prove that $\log(1 + \sqrt{1+x^2})$ is uniformly continuous?

Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous. My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = ...
5
votes
4answers
2k views

How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
5
votes
1answer
1k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
5
votes
2answers
444 views

Logarithm of a Markov Matrix

Start with a Markov matrix $\mathbf{M}$, whose elements are all between $0 \le \mathbf{M}_{ij} \le 1$ and each row sums to one. There is a natural connection with this matrix and the rate matrix ...
5
votes
3answers
335 views

Solving $\log _2(x-4) + \log _2(x+2) = 4$

Here is how I have worked it out so far: $\log _2(x-4)+\log(x+2)=4$ $\log _2((x-4)(x+2)) = 4$ $(x-4)(x+2)=2^4$ $(x-4)(x+2)=16$ How do I proceed from here? $x^2+2x-8 = 16$ $x^2+2x = 24$ ...
5
votes
2answers
256 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
5
votes
2answers
302 views

Definition of a logarithm

My question is as follows: Is the below a useful elementary way of dealing with negative arguments? If not, what is a better (elementary or not) way of dealing with negative arguments of the ...
5
votes
3answers
203 views

Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?

It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality: $$\int_{1}^{n} \log x \,dx \le ...
5
votes
1answer
108 views

Solutions for this logarithmic equation.

For which values of $k$ does the equation $\log_a(kx+3)+\log_a(x+1)=\log_a(2x+1)$ have one or more solutions in $x$? The logarithmic functions must have the restriction that the argument is ...
5
votes
2answers
69 views

Is $(\log(n))!$ a polynomially bounded function?

Is the following statement true? How would you prove it? i.e. Is it a polynomially bounded? $$ \lceil \lg(n) \rceil ! \in O(n^k) $$ How about $$ \lceil \lg \lg(n) \rceil ! \in O(n^k) $$ Thanks a ...
5
votes
2answers
105 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
5
votes
2answers
65 views

How to find the limit of $\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$

How to find the following limit: $$\lim_{n\to \infty} n(H(n) - \ln(n) - \gamma)$$ where $H(n) = 1 + \frac{1}{2} + \cdots + \frac{1}{n}$ is the $n^{th}$ harmonic number and $\gamma$ is the Euler ...
5
votes
3answers
193 views

Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $

Given a series of the type: $$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$ How does one evaluate it? Something I noticed was: $$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = ...
5
votes
1answer
258 views

Explain this code to compute $\log(1+x)$

It's well known that you need to take care when writing a function to compute $\log(1+x)$ when $x$ is small. Because of floating point roundoff, $1+x$ may have less precision than $x$, which can ...
5
votes
1answer
505 views

Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
5
votes
1answer
87 views

Irrational to power of itself is natural

I've been thinking about a natural number like $n$ so that $x^x=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some ...
5
votes
2answers
144 views

How to formally show that $f(z)$ is analytic at $z=0$?

Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$? I know that for small $z$ we have ...
5
votes
1answer
241 views

Can I build a program that will tell me if a real world data set looks linear, logarithmic, exponential etc?

I have a bunch of real world data sets and from manually plotting some of the data in graphs, I've discovered some data sets look pretty much logarithmic and some look linear, or exponential (and some ...
5
votes
3answers
63 views

Limit of logarithmic function using l'Hospital

How can I find the following limit: $$\lim_{x\rightarrow \infty}\frac{\ln(1+\alpha x)}{\ln(\ln(1+\text{e}^{\beta x}))}$$ where $\alpha, \ \beta \in \mathbb{R}^+$. My first guess was to use ...
5
votes
1answer
408 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and ...
5
votes
1answer
610 views

Inverse of the polylogarithm

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) ...
5
votes
1answer
121 views

Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem

I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$. Nagura uses the following definitions: $$\vartheta(x) = ...
5
votes
1answer
144 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
5
votes
0answers
801 views

Choosing the branch of a logarithm

The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
5
votes
1answer
257 views

Series involving Logs

I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g. $$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$ I realise that this comes from $x^y ...
4
votes
4answers
307 views

Clarify why all logarithms differ by a constant

One of the rules of logarithms that I've been taught is: $\log_b(x)$ is equivalent to $\frac{\log_k(x)}{\log_k(b)}$. Recently I've also seen another rule that says: $\log_b(x)$ is equivalent to ...
4
votes
7answers
644 views

Prove that $5/2 < e < 3$?

Prove that $$\dfrac{5}{2} < e < 3$$ By the definition of $\log$ and $\exp$, $$1 = \log(e) = \int_1^e \dfrac{1}{t} dt$$ where $e = \exp(1)$. So given that $e$ is unknown, how could I ...
4
votes
4answers
149 views

Differentiate $\log_{10}x$

My attempt: $\eqalign{ & \log_{10}x = {{\ln x} \over {\ln 10}} \cr & u = \ln x \cr & v = \ln 10 \cr & {{du} \over {dx}} = {1 \over x} \cr & {{dv} \over {dx}} ...
4
votes
3answers
510 views

Solving $xe^{-x}+2e^{-x}=0$

While I was studying my maths book, I came across this equation: $$ xe^{-x}+2e^{-x}=0 $$ I tried to solve it in different ways, but each time I break up some rule. My best try was this: Let's ...
4
votes
5answers
407 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
4
votes
5answers
73 views

What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?

I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem. $$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$ I'm sure ...
4
votes
6answers
160 views

How to calculate $\ln(x)$

I know only to calculate $\ln()$ using a calculator, but is there a way to calculate it without calculator: for example: $\ln(4)= ??$ as far as I know the only way to do so is to draw the graph of ...
4
votes
7answers
254 views

Solving $x^{\log(x)}=\frac{x^3}{100}$

How do I find the solution to: $$x^{\log(x)}=\frac{x^3}{100}$$ So I multiplied 100 both sides getting: $$100x^{\log(x)}=x^3$$ Now what should I do?
4
votes
4answers
266 views

Can anybody explain how these logarithms are transformed?

I'm learning for my algorithms exam and I can't derive two logarithm transformations: $ 3^{log_{4}(n)}=n^{log_{4}(3)} $ $ log_{3}(n)=log_{3}(e)*ln(n) $ I'm not very strong in logarithms, anybody ...
4
votes
8answers
277 views

Find $\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$ efficiently

I need to evaluate: $$\lim_{x \to 0} \left( \frac{\ln (\cos x)}{x\sqrt {1 + x} - x} \right)$$ Now, it looked to me like a classic L'Hospital's Rule case. Indeed, I used it (twice), but then things ...
4
votes
4answers
142 views

Solve these equations simultaneously

Solve these equations simultaneously: $$\eqalign{ & {8^y} = {4^{2x + 3}} \cr & {\log _2}y = {\log _2}x + 4 \cr} $$ I simplified them first: $\eqalign{ & {2^{3y}} = ...
4
votes
4answers
392 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 ...
4
votes
4answers
1k views

Find the domain of $f(x)=\ln(3x+2)$

Find the domain of $f(x)=\ln(3x+2)$ I can find domain, but is it the same for a $\log$ function? And also, do I have to rid the equation of the $\ln$ before I can find the domain? I'm really ...
4
votes
2answers
318 views

How can we solve: $\sqrt{x} - \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} - \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions? This question is inspired by my first question How ...
4
votes
3answers
761 views

How popular and used were logarithm tables?

I've heard that, for a time, logarithm tables "sold more than the Bible". Can someone produce some reliable documentation about how prevalent they were ? Would a common shopkeep have one ? Would a ...
4
votes
2answers
14k views

The difference between log and ln

$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$ Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the ...
4
votes
2answers
320 views

Double integral application

I need to determine $$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\frac{1}{1-y}dydx$$ I integrate in terms of the y component and obtained: $$\int_{0}^{1}\ln(\frac{1+\sqrt{x}}{1-\sqrt{x}})dx$$ Can ...
4
votes
2answers
191 views

Limit of logarithms without l'Hospital

This is my first post so I hope you forgive any formatting mistakes. This is a task out of a training exam, I may add that we have not yet introduced l'Hospital or derivatives. We have to determine ...
4
votes
2answers
515 views

Logarithm as limit

Wolfram's website lists this as a limit representation of the natural log: $$\ln{z} = \lim_{\omega \to \infty} \omega(z^{1/\omega} - 1)$$ Is there a quick proof of this? Thanks
4
votes
4answers
301 views

Does $\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y$?

I hit a snag whilst revising some log rules, could anyone confirm my suspicion: $$\log _b \left( x \right) = \log _b \left( y \right) \rightarrow x = y ?$$
4
votes
2answers
140 views

Solve $ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0$

I'm new to logarithms and I am having trouble solving this equation $$ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0.$$ How would I solve this? A step-by-step response would be appreciated. ...
4
votes
3answers
106 views

Why is $x^{\log_x n}=n$?

I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$. The answer is $17$, which is the argument of the ...
4
votes
2answers
199 views

Homework with logarithms

I'm stuck on continuing the next exercise: Considering: $\log_{c}a = 3$ $\log_{c}b = 4$ and: $$ y = \frac{a^{3}\sqrt{b \cdot c^{2}}}{2} $$ What's the value of $\log_{c}y$ ...