Questions related to real and complex logarithms.

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3
votes
4answers
820 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
55 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
83 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
153 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
191 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
101 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. $\log_{3x}(81)=2$ ...
39
votes
2answers
1k 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: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...
5
votes
2answers
148 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
69 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
51 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 ...
-3
votes
1answer
245 views

Solving $(2x-1)\ln5=\ln2 + x\ln3$ for $x$

Solve for the value of $x$: $$(2x-1)\ln5=\ln2 + x\ln3$$
4
votes
2answers
97 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 ...
16
votes
1answer
242 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
1k 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
141 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
113 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. ...
17
votes
1answer
320 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
148 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.
7
votes
4answers
2k 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
98 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
85 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
59 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
66 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
84 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!) - ...
64
votes
3answers
2k views

Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

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 ...
1
vote
0answers
39 views

What's the most straight forward way to show that a function is increasing?

I am trying to show that: $$\frac{2}{n}\log\Gamma\left(\frac{x}{2}\right) - \log\Gamma\left(\frac{x+n-1}{n}\right)$$ is an increasing function for $x \ge 5$ and $n > 2$ One way to do this would ...
3
votes
5answers
153 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
56 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
56 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 $x \ge 2000$: ...
0
votes
1answer
137 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
71 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
64 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
67 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
152 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} ...
2
votes
1answer
90 views

How to take cube of $\log$

What is the right way to solve $ \displaystyle (\log_2 n)^3 $ . I want to decompose it in plain $\log$ without exponent.
12
votes
3answers
463 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
4answers
138 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
76 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
55 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
96 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
72 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 ...
1
vote
2answers
432 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
398 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
70 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
191 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 ...
3
votes
2answers
217 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
170 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
2k 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 $X$ ...
2
votes
2answers
261 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
138 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 ...