Questions related to real and complex logarithms.

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2
votes
1answer
98 views

Calculate $I= \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}dx$

Please help me solve this: (level = high school) $$ \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}\,dx $$ Thanks
-1
votes
2answers
123 views

How to show $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$?

I was trying to solve $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ and I keep getting a partial answer of $x>4$ though answer key suggests a more expanded ...
0
votes
1answer
23 views

What is the minimum degree of x so that it is greater than or equal to ln(x)?

I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function n = ln(ln(x))/ln(x) but I'm not sure if that's gonna work. Thanks ...
1
vote
1answer
35 views

Why does the same inequality give different answers?

$\left(\log _2\left(x\right)-2\right)\left(\log _2\left(x\right)+1\right)<0$ has a solution $\frac{1}{2}<x<4$ But when we take the second part alone that is $\left(\log ...
1
vote
3answers
58 views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
2
votes
1answer
555 views

Simple proof Euler–Mascheroni $\gamma$ constant

I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \sum\nolimits_{k=1}^n \frac{1}{k} - \log(n)$ converges. At first I want to know if my answer is OK. ...
0
votes
2answers
59 views

Limit of a harmonic subseries minus “its” logarithm

$\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{3k-1} - \frac{1}{3}\ln(n)$ I think that inserting the other terms and then subtracting them would not help. I need just the ideea. Thank you.
0
votes
1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
0
votes
3answers
88 views

How can I solve the following equation? [closed]

$\log_2{\frac{x-3}{x+2}}≤0$ Thank you.
0
votes
2answers
35 views

Simultaneous log equations

I'm going through logarithms at the moment, and I can't solve this simultaneous equation: $$\log x - \log 2 = 2\log y$$ $$x - 5y + 2 = 0$$ I've tried substituting both $x$ and $y$ to no avail: ...
2
votes
2answers
46 views

solving equations with powers

Im trying to solve the equation $$3\cdot2^{-2/x} + 2\cdot9 ^{-1/x} = 5\cdot6^{-1/x }$$ So far I tried applying logaritmas but it didnt prove helpful...are there any other ways?
1
vote
1answer
23 views

Using Stirling's approximiation to show that $(\log(\log n))!$ is $O(n^k)$

I am trying to show the following: Prove, using Stirling's approximiation, that $(\log(\log n))!$ is $O(n^k)$ for some positive constant $k$. Stirling's approximation is $$n!=\sqrt{2\pi ...
1
vote
3answers
41 views

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$

If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$ My try: $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$ $log_e{(x-2y)^2} = log_e{xy}$ ...
3
votes
1answer
70 views

Why $\ln(1)\neq 2\pi ik$

Given that $e^{2\pi ik}=1$ for all $k \in \mathbb{Z}$, why isn't $\ln{e^{2\pi ik}}=2\pi ik$? On the other hand $\ln(1)=0$. What am I missing here?
8
votes
4answers
12k views

How do I find the base when Log is given

I'm trying to figure out how to calculate the base if: $$ \log_b 30 = 0.30290 $$ How do I find $b$ ? I've slaved over the Wikipedia page for logarithms, but I just don't get the mathematical ...
0
votes
1answer
48 views

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $

How to simplify this equation? $1+\sqrt {2^{2a_{n}+b_{n}+1}-16^{a_{n}}-4^{b_{n}}}=\log _{3}\left( a_{n}+b_{n}\right) $
2
votes
1answer
41 views

Name for a Logarithm Identity/Property

I came across a neat logarithm fact today: $\large n^{\log_bx} = x^{\log_bn}$ One simple proof is: $\large \log_bx\cdot \log_bn=\log_bx\cdot \log_bn$ $\large \Rightarrow ...
2
votes
2answers
81 views

Explanation of this inequality

Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ? I can't see why the inequality is true. I ...
0
votes
3answers
62 views

Solve logarithmic equation: $2\log_7 (x+2) - \log_7 (3x+10) = 0$ [closed]

Please, can someone check if this is the right answer $$x= -2 \pm \sqrt{3x + 10}$$ Thank you.
25
votes
3answers
810 views

If there are entire $G_k$s such that $f=\exp\circ\exp\circ\cdots \circ\exp\circ G_k$ ($k$ times), must $f$ be constant?

I am a French guest and I hope that my English isn't too bad... So here is my issue: I consider an entire function $f$ which satisfies the following property for each complex number $z\in ...
0
votes
2answers
28 views

How to interpret the difference in log points

How can we interpret the difference between two log points? Is it correct to interpret this difference in percentage points? Thanks. Marko
0
votes
1answer
18 views

Solving Logarithms involving ceiling function

I need to solve the equation $\lceil \log_B(M) \rceil = S$ for $B$ when $M$ and $S$ are known, $M$ and $S$ are integers, and $B < M$. Were the ceiling function not there, it would be trivial, ...
4
votes
5answers
70 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 ...
2
votes
2answers
87 views

How do I simplify $\log (1/\sqrt{1000})$?

How do I simplify $\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right)$? What I have done so far: 1) Used the difference property of logarithms $$\log ...
2
votes
3answers
108 views

A series converging (or not) to $\ln 2$

I have come across the following series, which I suspect converges to $\ln 2$: $$\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}.$$ I could not derive this series from some of the standard ...
3
votes
4answers
56 views

A general definition of Entropy (i.e. may or may not be expectation of the Log of the probabilities) [closed]

Entropy may be defined as Entropy = Σ G(p(x)) Where 'G' is any function that goes asymptotically to plus infinity as it approaches zero from the positive side and is monotonic between 0 and 1 ...
2
votes
4answers
20k views

Find the domain and range of a logarithmic function?

I need help finding the domain and range of logarithm functions. For example, what is the domain and range of $y=\log(x-3)$?
36
votes
1answer
1k 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
2answers
64 views

Proof the expession $\log_{12}{18}*log_{24}{54} + 5(\log_{12}{18}-log_{24}{54})=1$

I am trying to proof the following expression (without a calculator of course). $\log_{12}{18}*\log_{24}{54} + 5(\log_{12}{18}-\log_{24}{54})=1$ I know this isn't a difficult task but it's just ...
0
votes
2answers
441 views

merge sort vs insertion sort time complexity

How do I solve exercise 1.2-2 from Introduction to Algorithms 3rd Edition, Author: Thomas H. Cormen Would I need to set both sides equal to each other and solve for n?
1
vote
2answers
49 views

Definition of $a^b$ for complex numbers

Problem statement Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$ $(i)$ Verify that if $b \in ...
2
votes
1answer
36 views

How to solve this logarithmic equation?

I want to solve this equation: $$8n^2 = 64n\log_{\ 2}(n)$$ After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to ...
2
votes
2answers
49 views

I need help on the process of solving this derivative.

How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$ I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
6
votes
3answers
181 views

Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$

I'm look for a closed form evaluation of the following improper definite integral involving logarithms: $$\begin{align} I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\ ...
0
votes
2answers
41 views

Help me solve this…

Assuming $a=\log 2$ and $b=\log 3$ (log is the base 10 logarithm). I have to find $\log_5 288$. How can I do this? Edit: I've tried transforming $\log2$ to $\frac{\log_5 2}{\log_5 10}$ and same for ...
0
votes
0answers
24 views

Using math functions to time finales of a fireworks show

This year, I have the honor of programming two finales for a fireworks show. I want to use math. I suspect that I should use a function such as square root or log to specify the decreasing pause ...
0
votes
1answer
11 views

Consumption change calculation

I want to calculate yearly consumption change according to the following formula: $$C_{t+1}=C_{t}e^{x_{t}}$$ I need to calculate ${x_{t}}$. I have the consumption data $C_{t+1}$ and $C_{t}$.
7
votes
3answers
196 views

Is $ln(x)$ ever greater than $x$

Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement? Just wondering for some convergence related thing
0
votes
2answers
51 views

Help me to solve math homework on logarithmic

How to solve this math home work? Please help.. What is the value of $\log \left(\dfrac{i\pi}{2}\right)$ ? I got to know the answer is "$\dfrac{i\pi}{2}$", but don't know how to solve it. Please ...
2
votes
1answer
30 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
-3
votes
2answers
47 views

How to get this answer [closed]

Anyone help me solve this question $$\ln u + 2 \ln(1-u) - 2 \ln(1+u) = 2 \ln x + \ln c$$ I have the answer as $\frac{x y}{ (x^2 - y^2)^2} =c$, but I cant figure out how get this answer.
4
votes
2answers
107 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I ...
2
votes
1answer
71 views

Checking derivation of y = a^x

Can you tell me if there are any flaws with this derivation of $y = a^x$... The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
2answers
39 views

Avoiding substraction for finite difference with log and exp

I want to approximate the derivative of f(x) Finite difference $f'(x) \approx \frac{f(x+h)-f(x)}{h}$ I was taught that the error from the substraction is blown up for small h. This I can verify ...
0
votes
1answer
44 views

Is there a property for log(n)/n?

I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part: $$\frac{1}{8} = ...
0
votes
2answers
37 views

Logorithms on a first level learning

Solve log$_{5x-1}$ $4$ $=$ $1/3$ $(5x-1)^{1/3}$=4 $((5x-1)^{1/3})^3$ = $4^3$ $5x-1=64$ $5x=65$ $13$ I am not sure where to go with this. I learned some things about logs before my class ended ...
7
votes
1answer
270 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?
0
votes
1answer
27 views

Mathematics - geometric progression question

If $a$, $b$ and $c$ are in geometric progression, then what are $\log_ax$, $\log_bx$ and $\log_cx$ in? What I did: I substituted values for $x, a, b$ and $c$ and tried to solve it further. What I ...
2
votes
1answer
57 views

What does this log notation mean?

Can someone please explain what $^2\log x$ means? Is it the same as saying $\log x^2$ or is it something completely different? Here is an image of it as an example:
68
votes
3answers
2k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...