Questions related to real and complex logarithms.
3
votes
3answers
50 views
If $z_n \to z$ then $(1+z_n/n)^n \to e^z$
We are dealing with $z \in \mathbb{C}$.
I know that
$$
\left(1+ \frac{z}{n} \right)^n \to e^{z}
$$
as $n \to \infty$. So intuitively if $z_n \to z$ then we should have
$$
\left(1+ \frac{z_n}{n} ...
5
votes
3answers
221 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
3
votes
3answers
88 views
What math will I need in order to learn Microsoft's UProve?
I'm studying Microsoft's UProve (independent studies at 35 years old) and forget most of the Math I learned in college.
I intend to proceed and learn the contents of this chapter of this book but can ...
0
votes
1answer
45 views
How do I create an equation that decelerates past a certain value?
Apologies for my lack of pure maths, I am a programmer!
I currently have an equation in code that states that if a number goes below a certain value (in my case, 0.7) then the difference is dampened:
...
0
votes
1answer
40 views
How would I evaluate $y=e^{2\frac{\ln2}{3}}+e^{-\frac{\ln2}{3}}$?
$$y=e^{2\frac{\ln2}{3}}+e^{-\frac{\ln2}{3}}$$
I am not sure how I would go about evaluating this. I have tried rewriting the expression by splitting up the $e$ and the exponents but it just seemed to ...
2
votes
3answers
42 views
Logarithm rules, which one has priority? $\ln2e^{2x}$
$$\ln2e^{2x}$$
Here are the two results I came up with:
$$2x(\ln2e)$$
$$2x(\ln2+\ln e)$$
$$2x(\ln2 + 1)$$
$$2x\ln2+2x$$
and
$$\ln2+\ln e^{2x}$$
$$\ln2+2x\ln e$$
I am sort of leaning towards the ...
1
vote
2answers
37 views
Equation with Logarithm
I want to solve the following equation:
$$3^x3^{x-1} = 243.$$
My approach is the following:
$3^{2x-1} = 243$ then:
$(2x-1)\cdot\log3 = \log 243$ and then:
$x = (\frac{\log243}{\log3}+1)/2$
Is ...
0
votes
2answers
58 views
Mental Math - Estimating Logarithms
How can we estimate logarithms with different bases? Take $\log_2 10$ ($1\over\log_{10}2$$\approx3.32192809$) for example. If we convert $10$ to binary, we get $1010_2$. So $\log_21010_2$ can clearly ...
7
votes
3answers
189 views
Broken Calculator: only certain unary functions work.
I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic ...
1
vote
1answer
32 views
Equation with Logarithm
Given is the equation:
$$\log_x3+\log_x12 = 2$$
How do I solve it? My idea was to use the formula $\log_a(b) = \frac{\ln b}{\ln a}$ but that does not seem to help here..
1
vote
1answer
31 views
Intersection of two functions, logarithms
I have to calculate the intersections of the two following functions:
(i) f(x) = $3^x$ and $g(x) = 2^{-x}$
(ii) f(x) = $e^{-x}$ and $g(x) = 2e^x$
and I must do a mistake somewhere but I don't know ...
0
votes
0answers
16 views
Question concerning expansion of the log function.
I'll get straight to it.
$\ln(x)=\int\frac{1}{x}dx
=\int\frac{1}{1-(1-x)}dx$
And $\frac{1}{1-(1-x)}=\sum_{n=o}^{\infty}(1-x)^n$
Am I correct so far? Because on wikipedia, the series ...
3
votes
2answers
44 views
Product rule for logarithms works on any non-zero value?
The product rule for logarithms states that:
$$\log_b M + \log_b N = \log_b (M\cdot N)$$
Most sources that I've found* state that this rule only applies when $M$ and $N$ are positive. It's true that ...
1
vote
3answers
63 views
Expand into power series $f(x)=\log(x+\sqrt{1+x^2})$
As in the topic, I am also supposed to find the radius of convergence.
My solution: $$\log(x+\sqrt{1+x^2})=\log \left ( x(1+\sqrt{\frac{1}{x^2}+1})\right )=\log(x)+\log(1+\sqrt{\frac{1}{x^2}+1})$$Now ...
1
vote
1answer
56 views
Derive the PDF of the log-normal distribution?
If $X \sim N(0,1)$ and $Y = e^X$, find the PDF of $Y$ using the two methods:
(i) Find the CDF of of $Y$ and then differentiate. Use the notation $\Phi(x)$ and $\phi(x)$ for the CDF and PDF of $X$ ...
0
votes
2answers
42 views
Definition of logarithm in complex domain
My first question is:
What is the proper definition of logarithmic function $f(z)=\ln{z}$.
where $z\in \mathbb{C}$.
quoting Wikipedia.
a complex logarithm function is an "inverse" of the ...
3
votes
2answers
85 views
Meaning of $\log$
If you write $\log{x}$ rather than ${\log_a{x}}$ for some base $a$, does it have a particular meaning? Sometimes I see people leave off the base by mistake when posting questions and it seems from the ...
5
votes
1answer
98 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:
$$\vartheta(x) = ...
1
vote
1answer
65 views
Comparing rates of change: which function increases faster?
I am comparing two functions for $x \ge 1$:
$$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$
$$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
0
votes
2answers
48 views
Quick logarithm calculation
In coming up with an algorithm for finding log (10) base 2, these are my thoughts. I wanted to know if this makes sense and how could I truly make it more efficient. The requirements are strictly not ...
3
votes
1answer
68 views
What is the closed formula for the following summation?
Is there any closed formula for the following summation?
$$\sum_{k=2}^n \frac{1}{\log_2(k)}$$
1
vote
0answers
24 views
transforming a straight band into a logarithmic spiral
I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral.
If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
1
vote
1answer
29 views
simple logarithms with exponents
Note: I am using log base 10 and I am trying to rewrite the equation using exponents instead of logs.
Here is what I have and I am wondering if I did it correctly (if not how am I suspose to solve ...
0
votes
1answer
27 views
1
vote
1answer
45 views
Solving basic exponential equation with logs
I am having trouble with this grade 12 pre-calc question that I am sure will be elementary to most of you. I understand most of it but I do not understand one of the steps.
These are the steps in my ...
1
vote
1answer
46 views
Is this a valid way to evaluate a function based on factorials? What would be a better way?
I am working on the following factorial function:
$$f(x) = [\ln(\lfloor\frac{x}{11}\rfloor!) - \ln(\lfloor\frac{x}{12}\rfloor!) - \ln(\lfloor\frac{x}{132}\rfloor!)] + ...
5
votes
1answer
67 views
How to compute the asymptotic growth of $\binom{n}{\log n}$?
I'm interested with tight bounds for: $$f(n)={n\choose{\log{n}}}$$
It sounds like it's something simple, but I can't get a nice expression I can use.
Any ideas on how to do this?
1
vote
2answers
46 views
Discrete Logarithm
If $p$ is a prime and $a,b$ are integers not divisible by $p$ such that $a^x \equiv b \pmod p$ with $0 ≤ x < o_p(a)$, then we define $x = L_a(b)$ and say $x$ is the discrete logarithm of $b$ ...
1
vote
1answer
40 views
Denesting Logarithmic expressions
$\log_7(\log_2(3)) + \log_7(\log_5(6)) + \log_7(\log_{11}(1/2)) = \log_7(-1) + \log_7(\log_5(3)) + \log_7(\log_{11}(6))$
This can only be simplified by using the sum to product rule and noticing that ...
0
votes
1answer
68 views
Proving that a specific gamma function is a guaranteed lower bound for a factorial function
In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that:
$$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$
I have ...
1
vote
2answers
29 views
Logarithm of a Gram matrix
Given a Gram matrix $K$, we are interested in calculating its matrix logarithm $\log(K)$, and in particular, to relate minus this logarithm to the Laplacian of a graph.
We have noticed that ...
1
vote
1answer
22 views
Finding solutions to $\frac{2 t }{x}= B_r \log B_r + B_s + B_s x \log(B_s x)$
Given $B_r$, $B_s$, $t$ being constants, and $x$ being a variable $0\leq x\leq 1$ how can I solve this equation?
$$\frac{2 t }{x}= B_r \log B_r + B_s + B_s x \log(B_s x)$$
If i plot the two ...
1
vote
1answer
27 views
Looking for help understanding the asymptotic expansion of the digamma function
I was recently given an example using this asymptotic expansion of the digamma function where:
$$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$
Here's the ...
4
votes
3answers
100 views
To find the logarithm of $1728$ to the base $2 \sqrt{3}$
Find the logarithm of: $1728$ to base $2\sqrt{3}$.
Let, $\log_{2\sqrt{3}} 1728 = y$, then
$$\begin{align} (2\sqrt{3})^y &= 1728\\
2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
0
votes
1answer
37 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
1
vote
0answers
28 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
4
votes
1answer
109 views
Need help understanding if a function is increasing or decreasing
I am working on understanding the following function:
$$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
0
votes
4answers
129 views
Why aren't logarithms defined for negative $x$?
Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
2
votes
1answer
43 views
The rate of increase of the Gamma Function over real numbers
If
$$ x_1 > x_2 > 0$$
and $$\Delta{x}>0$$
does it follow that:
$$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$
Would it be enough to show ...
2
votes
4answers
122 views
How do i find the inverse of: $f(x) = {2^{x - 1}} - 3$
$f(x) = 2^{x - 1} - 3$
My approach:
Take logs to base 2:
$ = \log_2 \left( x - 1 \right) - \log_2 \left( 2^3 \right)$
$ = \log_2 \left( {x - 1} \over {2^3} \right)$
This isn't the answer in the ...
1
vote
2answers
36 views
Prove that $\ln$ has an inverse function
For $x$ in $(0, \infty)$ let $\ln(x) = \int_{1}^{x}\frac{1}{t}dt$.
Prove that $\ln$ has an inverse function
My book does not really go into how to prove something has an inverse, besides it needing ...
1
vote
0answers
101 views
Using the gamma function as an upper and lower bound to the logarithm of a factorial function.
I am trying to find an upper and lower bound for the following function:
$$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$
where
...
5
votes
4answers
322 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
0
votes
2answers
68 views
Why I can't calculate $0*log(0)$ but can $log(0^0)$
I got this doubt after some difficult in programming. In a part of code, i had to calculate:
$$
x = 0 * Log(0) \\ x = 0*-Inf
$$
and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
1
vote
1answer
42 views
Understanding the upper and lower bounds of the error estimate in Stirling's Approximation
Based on the Wikipedia article on Stirling Approximation:
$n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$
where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$
How would this ...
20
votes
0answers
413 views
Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
0
votes
1answer
14 views
Some number to an exponent of a log
I need to simplify an expression. I am currently working on the following problem (I apologize in advance for formatting, I'm not sure how to use it on Stack Exchange):
$81^{(\log_{3}N)+(log_{9}N)}$
...
2
votes
1answer
35 views
How to compute a product of logarithms?
I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on -
What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$
I did some searching, and found ...
2
votes
0answers
29 views
Discrete logarithm - strange polynomials
If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
