Questions related to real and complex logarithms.
2
votes
1answer
42 views
Natural logarithm
Can someone please suggest how one proves:
$(1+2x)\ln(1+\frac{1}{x}) -2 >0$ where $x>0$.
I plotted the function in a program and the inequality should be correct.
10
votes
3answers
113 views
Closed form for n-th anti-derivative of $\log x$
Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
1
vote
2answers
24 views
simplifying equation with logs
I have the following equation:
I would like to solve this for Ze. I have found the same equation expressed in terms of Ze in another paper:
I can't get my head around how this works. This is my ...
3
votes
3answers
59 views
logs Challenge between two students >>be smart
two student were given the equation $2^{4x+6} = 3^{6x-3}$
1.steve rearranged to get $2^{4x+6} - 3^{6x-3} =0$
then wrote $\log (2^{4x+6} - 3^{6x-3}) = \log0$
are these legal steps ? if not explain ...
1
vote
1answer
66 views
How do I divide a set of data samples which follow a logarithmic distribution?
I'm working for the first time with Logarithmic distribution. I have a set of samples which follow logarithmic distribution. I extracted the maximum and the minimum values from the set and defined the ...
1
vote
2answers
168 views
Taylor series expansion of base 2 logarithms
Sorry for the noob question, but I've been hitting my head against the wall on this for a while.
I am looking for a Taylor series expansion of a logarithm other than the natural logarithm $ln(x)$. It ...
21
votes
0answers
417 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 $x \ge 1331$, ...
1
vote
0answers
40 views
simplification of a natural log of a trigonometric function
hope you are all well.
I am having a bit of a mental block, I am wondering if it is possible to simplify the following expression:
$$k\cos X \cdot 4\ln(\cos X)$$
where $k$ is a constant and $X$ ...
1
vote
4answers
41 views
Logarithm question involving different base [closed]
Calculate the values of $z$ for which $\log_3 z = 4\log_z3$.
1
vote
4answers
153 views
Examples of logs with other bases than 10
From a teaching perspective, sometimes it can be difficult to explain how logarithms work in Mathematics. I came to the point where I tried to explain binary and hexadecimal to someone who did not ...
0
votes
3answers
54 views
On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity.
Trying to solve similar type equation to this.
On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity. According to the diagram, vinegar is how many times as ...
2
votes
3answers
76 views
$a^b = c$, is it possible to express $b$ without logarithms?
$ a^b = c $
is it possible to express b without logarithms?
12
votes
1answer
91 views
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
I encountered this integral in my calculations:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
0
votes
0answers
62 views
Floor of log equation $S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+1}-10}{9}$
I must find 'x' and I don't know how to solve the following equation.
Does it have a solution? How can I solve it?
$$
S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - ...
3
votes
4answers
256 views
Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$
Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$
How to prove without a computer?
0
votes
2answers
43 views
Solve a simple equation with log in it
I'm stuck with solving this equation,
$$2 \log x = \log 9 $$
This is how far I made it:
\begin{align}
\log x &= \log 4,5 \\
x &= ?
\end{align}
I'm a beginner at logarithms so I appreciate ...
2
votes
5answers
92 views
How to write in $2^x=5$ in logarithmic form?
How do I write:
$$2^x = 5$$
In a logarithmic form?
I've looked for a solution for some time now, so I decided to try here.
3
votes
2answers
48 views
Separating $\frac{1}{1-x^2}$ into multiple terms
I'm working through an example that contains the following steps:
$$\int\frac{1}{1-x^2}dx$$
$$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$
$$\ldots$$
$$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$
I ...
4
votes
4answers
74 views
Prove $\ln{(\frac {x}{y})} = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
Prove $\ln (\frac{x}{y}) = \ln{x} - \ln{y}$ using the definition $\int_1^x\frac{1}{t}dt = \ln{x}$.
I am able to prove $\ln{xy} = \ln{x} + \ln{y}$, and $\ln{x^r} = r\ln{x}$, but with this one, I am ...
2
votes
2answers
30 views
rounding up to nearest square
Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner?
ie.
$2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. ...
4
votes
7answers
111 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?
0
votes
2answers
53 views
Logarithmic equation. Need to know if i am teaching right
Two of my friends is studying for a test. They asked me about a simple question. But they told me that i was wrong on a question. I could be wrong. But i need you guys to make sure that they learn the ...
2
votes
0answers
54 views
Convexity of polylogarithms
I want to prove the following proposition:
The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.
And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
5
votes
5answers
122 views
Prove that $\log X < X$ for all $X > 0$
I'm working through Data Structures and Algorithm Analysis in C++, 2nd Ed, and problem 1.7 asks us to prove that $\log X < X$ for all $X > 0$.
However, unless I'm missing something, this can't ...
3
votes
4answers
107 views
Integrate by parts: $\int \ln (2x + 1) \, dx$
$$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x ...
3
votes
1answer
122 views
How to prove $\left\|\ln\left(e^{iH_1}e^{iH_2}\right)\right\|\leq\left\|H_1\right\|+\left\|H_2\right\|$?
Let $H_1$ and $H_2$ denote arbitrary Hermitian operators (finite dimensional) and let $\left\|\ldots\right\|$ denote the usual operator norm. I conjecture that
$$
...
0
votes
1answer
25 views
Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically
$\eqalign{
& y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr
& \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr
& {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
1
vote
3answers
47 views
Evaluating a limit with variable in the exponent
For
$$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$
I have to use the L'Hospital"s rule, right? So I get:
$$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$
And ...
0
votes
1answer
27 views
1
vote
2answers
41 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
35 views
Logarithmic function
Solve for x;
$\log_{12}x=\frac{1}{2}\log_{12}9+\frac{1}{3}\log_{12}27$
The only thing throwing me off is the one third and one half, which my book does not say how to fix.
3
votes
2answers
52 views
Log problem, $u$ substitution the only way?
Okay so basically I want to know if you can solve this log equation without the use of u substitution:
$${\log_4{\log_3{x}}} = 1$$
I believe that u substitution is the only way to solve this ...
3
votes
4answers
43 views
Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$
What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming ...
4
votes
4answers
125 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}} ...
14
votes
1answer
145 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
1
vote
2answers
67 views
Pre Calculus Math Equation With Logarithms
Please Help me with this I think i figured out question 1... but I get no solution...
please help me start number 2 or if you can show full solution that be sick thanks.
$\log_{3x}(81)=2$
...
5
votes
2answers
91 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 ...
4
votes
0answers
55 views
Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$
I've been told that the approach below will not work.
I would be interested if someone could help me to understand what will go wrong.
Let:
$$\psi(x) = \sum\limits_{p^k \le x} \ln p$$
So that (see ...
0
votes
1answer
38 views
Looking for suggestions on how to proceed with showing that:
for $x \ge 2863:$
$$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$
I've written a java application which ...
4
votes
2answers
114 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
1answer
111 views
5
votes
4answers
116 views
Summation of logs
Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
9
votes
1answer
69 views
Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$
Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
0
votes
1answer
34 views
A matrix has a real logarithm if it has a positive spectrum.
The title is a proposition I read in my notes that's left with no proof. Where can I read one?
1
vote
1answer
38 views
All the logarithms of a non-singular matrix.
I'm reading some notes on dynamical systems that talk about matrix logarithms with little to no detail on the subject. I read the wikipedia article and others on the internet, but not all is clear.
...
15
votes
1answer
206 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
4
votes
4answers
104 views
How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?
How do you find
$$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$
I know it's $-1$, but I had to plot it.
34
votes
3answers
647 views
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$
Inadvertently, I find this interesting inequality,But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find ...
14
votes
5answers
501 views
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
4
votes
0answers
78 views
Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?
I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
