Questions related to real and complex logarithms.
1
vote
4answers
22 views
Logarithm question involving different base
Calculate the values of $z$ for which $\log_3 z = 4\log_z3$.
2
votes
3answers
74 views
$a^b = c$, is it possible to express $b$ without logarithms?
$ a^b = c $
is it possible to express b without logarithms?
11
votes
1answer
67 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:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
0
votes
3answers
50 views
On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity.
Trying to solve similar type equation to this.
On the pH scale, each unit change in pH represents a tenfold increase in acidity or alkalinity. According to the diagram, vinegar is how many times as ...
0
votes
0answers
53 views
Floor of log equation $S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - \frac{10^{\lfloor\log_{10}(x)\rfloor+1}-10}{9}$
I must find 'x' and I don't know how to solve the following equation.
Does it have a solution? How can I solve it?
$$
S=\left(\lfloor\log_{10}(x)\rfloor+1\right)x - ...
8
votes
1answer
63 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 $\log x$?
17
votes
0answers
82 views
Closed form for $\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
...
2
votes
5answers
88 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.
4
votes
4answers
73 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
29 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
104 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
47 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 ...
0
votes
2answers
40 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 ...
4
votes
4answers
101 views
Integrate by parts: $\int \ln (2x + 1) \, dx$
$$\eqalign{
& \int \ln (2x + 1) \, dx \cr
& u = \ln (2x + 1) \cr
& v = x \cr
& {du \over dx} = {2 \over 2x + 1} \cr
& {dv \over dx} = 1 \cr
& \int \ln (2x ...
0
votes
1answer
24 views
Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically
$\eqalign{
& y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr
& \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr
& {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
1
vote
3answers
46 views
Evaluating a limit with variable in the exponent
For
$$\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)$$
And ...
4
votes
4answers
122 views
Differentiate $\log_{10}x$
My attempt:
$\eqalign{
& \log_{10}x = {{\ln x} \over {\ln 10}} \cr
& u = \ln x \cr
& v = \ln 10 \cr
& {{du} \over {dx}} = {1 \over x} \cr
& {{dv} \over {dx}} ...
3
votes
4answers
41 views
Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$
What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$
This is what I got out of the equation so far. the alternate form assuming ...
1
vote
2answers
64 views
Pre Calculus Math Equation With Logarithms
Please Help me with this I think i figured out question 1... but I get no solution...
please help me start number 2 or if you can show full solution that be sick thanks.
$\log_{3x}(81)=2$
...
13
votes
1answer
125 views
Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$
Please help me to find a closed form for the following integral:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
5
votes
2answers
90 views
How to formally show that $f(z)$ is analytic at $z=0$?
Let $z$ be a complex number. Let $$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$$
So that (see ...
0
votes
1answer
37 views
Looking for suggestions on how to proceed with showing that:
for $x \ge 2863:$
$$\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)$$
I've written a java application which ...
-4
votes
1answer
107 views
2
votes
0answers
54 views
Convexity of polylogarithms
I want to prove the following proposition:
The function $w\to (-Li_{5/2}(-e^w))^{2/5}$ is convex on $\mathbb R$.
And, as I think, the same is true for the function $w\to (-Li_{p}(-e^w))^{1/p}$ for ...
8
votes
1answer
64 views
Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$
Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
5
votes
4answers
113 views
Summation of logs
Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
0
votes
1answer
33 views
A matrix has a real logarithm if it has a positive spectrum.
The title is a proposition I read in my notes that's left with no proof. Where can I read one?
1
vote
1answer
36 views
All the logarithms of a non-singular matrix.
I'm reading some notes on dynamical systems that talk about matrix logarithms with little to no detail on the subject. I read the wikipedia article and others on the internet, but not all is clear.
...
15
votes
1answer
204 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
4
votes
4answers
103 views
How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?
How do you find
$$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$
I know it's $-1$, but I had to plot it.
5
votes
5answers
116 views
Prove that $\log X < X$ for all $X > 0$
I'm working through Data Structures and Algorithm Analysis in C++, 2nd Ed, and problem 1.7 asks us to prove that $\log X < X$ for all $X > 0$.
However, unless I'm missing something, this can't ...
4
votes
0answers
78 views
Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?
I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
0
votes
0answers
58 views
Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.
In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that:
$$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
41 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
0
votes
1answer
25 views
How do you evaluate an inequality that involves logarithms of factorials?
For $x > 1$, $n > 2$ with $2 \mid x+1$ and $n \mid x+1$, does it then follow that:
$$\log(\lfloor\frac{x}{2}\rfloor!) - \log(\lfloor\frac{x}{n}\rfloor!) \le \log(\lfloor\frac{x+1}{2}\rfloor!) - ...
33
votes
3answers
611 views
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$
Inadvertently, I find this interesting inequality,But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find ...
0
votes
0answers
31 views
Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?
The answer seems to be yes.
Here's my reasoning.
Let $\{x\} = x - \lfloor{x}\rfloor$
Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$
$$\log(\lfloor\frac{x}{2}\rfloor!) = ...
1
vote
0answers
27 views
What's the most straight forward way to show that a function is increasing?
I am trying to show that:
$$\frac{2}{n}\log\Gamma(\frac{x}{2}) - \log\Gamma(\frac{x+n-1}{n})$$
is an increasing function for $x \ge 5$ and $n > 2$
One way to do this would be to show that ...
3
votes
6answers
127 views
Solve for $x$: question on logarithms.
The question:
$$\log_3 x \cdot \log_4 x \cdot \log_5 x = \log_3 x \cdot \log_4 x \cdot \log_5 x \cdot \log_5 x \cdot \log_4 x \cdot \log_3 x$$
My mother who's a math teacher was asked this by one of ...
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.
4
votes
1answer
46 views
Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$
I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning.
With Lemma 1, he establishes for $x \ge 2000$:
...
-3
votes
0answers
42 views
0
votes
1answer
55 views
Smallest Mersenne prime with 100 million digits?
As some of you are probably aware, the Great Internet Mersenne Prime Search (GIMPS) is managing the search for the largest Mersenne primes of the form $M_p=2^p-1$, where $p$ is itself prime (GIMPS ...
1
vote
4answers
48 views
Find the equation of the tangent to the curve $y = {2^x} + {2^{ - x}}$ at the point $(2,4{1 \over 4})$
$\eqalign{
& y = {2^x} + {2^{ - x}} \cr
& \ln y = x\ln 2 - x\ln 2 \cr
& \ln y = 0 \cr
& {1 \over y}{{dy} \over {dx}} = 0 \cr
& {{dy} \over {dx}} = 0 \cr} $
I've ...
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 ...
1
vote
2answers
43 views
Mean and variance of $\ln(u)$
Suppose $U$ follows $U(0.1)$.
1) find the mean and vairance of $\ln(u)$.
Question:
I wish to confirm the 1st part of the proof. Are these steps correct?
CDF of $Y = P(Y \leq y) = P(\ln{U} \leq y) ...
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
$$
\left(1+ \frac{z_n}{n} ...
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\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 ...
