Questions related to real and complex logarithms.

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2answers
230 views

Why does the log-normal probability density function have that extra “x”?

For a random variable $X \sim N(\mu, \sigma^2)$, the probability density function is $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}$$ ...
0
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1answer
48 views

How do we solve for $n$?

Asymptotic complexity gives an idea of how rapidly the space/time requirements grow as problem size increases. • Suppose we have a computing device that can execute 1000 complex operations per second....
3
votes
2answers
822 views

Log or Antilog tables, which ones are more useful?

I'm trying to make a Log or Antilog table small enough to fit in the back of a wallet calendar (or a business card). My intend is to build a mathematically useful gift that can be used by anybody ...
5
votes
3answers
107 views

Closed form for the partial sum $\sum\limits_{k = 1}^n \frac{\ln k}k$

I'd like to find a closed form for this partial sum: $$\sum\limits_{k = 1}^n \frac{\ln k}k$$ Using the properties of the logarithms, I converted the above into $$\ln\left(\prod_{k = 1}^n k^{1/k}\...
0
votes
3answers
72 views

Integrate a power of logarithm [closed]

Is there some way, how to solve this problem? $$ \int \ln^n(x) dx \text{, where } n \in \mathbb{N} $$ I really don't know, what to do with $n$.
-2
votes
4answers
68 views

Logarithm and trigonometry

Is $\ln (\sin x-\cos x)$ equal to $\ln (\cos x-\sin x)$? So I did a integral problem but the answer is not same the answer given. I'm given this question $\int (\frac{2}{1-\tan x})dx$ So I got ...
0
votes
1answer
89 views

Iterative Logarithm in Recurrence Relation?

Anyone Could describe me How we can solve this recurrence relation? $T(n) = T(\log n) + O(1)$ $T(1) = 1$ a) $O(\log n)$ b) $ O (\log^* n) $ c) $ O (\log^2 n) $ d) $ O (n / \log n) $ Our TA ...
1
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0answers
26 views

Minimum of the difference of two logarithms

I am trying to find an analytical expression of the minimum of $$ f_n(x) = \frac{2x}{n^2+n}\log(x) - \frac{2x+2}{n^2+3n+2}\log(x+1) $$ when $x\in [1;n]$ I used to think from graphing it that this ...
2
votes
1answer
100 views

Show $\log(1+x)$ is not a contraction mapping

Show $F:[0,\infty] \to [0,\infty]$, $F(x) = \log(1+x)$ is not a contraction mapping. Attempt: Assume $F$ is a contraction mapping, then we have that $\forall x,y \in [0,\infty)$, $|F(x) - F(y) | \...
1
vote
1answer
56 views

Natural logarithm equation, beginner stage

I am learning about natural logarithms and this is the first equation i must solve: $$ 30 - 23 e^{-0.027x} > 20 $$ Could somebody explain what i should do to solve this and other equations like ...
3
votes
3answers
172 views

Show that if $1> x>0$, then $x-1 ≥ \ln(x) ≥ 1−1/x$

Show that if $1> x>0$, then $x-1 ≥ \ln(x) ≥ 1−1/x$. I know the is using the MVT I can proof it for $x> 1$ but I don't understand how to proof for $x > 0$ .
2
votes
3answers
120 views

What is the correct integral of $\frac{1}{x}$?

I understand that the graphs of $\log(x)$ and $\ln(x)$ both have derivatives (changes in slope) that follow the pattern of: $$\frac{d}{dx}\log_{b}x= \frac{1}{(x\ln(b))}$$ However, depending on the ...
1
vote
3answers
37 views

Differentiating this problem $\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}$

How does one differentiate the function $$y(t)=\frac{2t^{3/2}}{\ln(2t^{3/2}+1)}.$$ I am still tying to understand MathJaX and not sure what is wrong with the expression. Anyways, How do I start/...
1
vote
3answers
43 views

$N =\sum_{k = 1}^{1000}k(\lceil\log_{\sqrt{2}}k\rceil-\lfloor\log_{\sqrt{2}}k\rfloor). $

Find $N$ for $$N =\sum_{k = 1}^{1000}k\left(\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor\right)\;.$$ How could you solve this problem? Are there sigma rules or ...
0
votes
1answer
61 views

Strange log scale on a plot. How do I read this?

Doing an assignment with a strange log-log data plot. You'll notice that there at 14 segments per cycle, and they are not spaced as usual. Note the last 4 segments break the pattern of reduced spacing....
1
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3answers
19 views

Let $ f(x)= ( \log_e x) ^2 $ and (Integration by parts. Comparing integrals of different limits )

Let $ f(x)=( \log_e x) ^2 $ and let $ I_1= \int_{2}^{12} f(x) dx $ , $ I_2= \int_{5}^{15} f(x) dx $ and $ \int_{8}^{18} f(x) dx$ Then which of the following is true? (A)$I_3 <I_1 < I_2 $ (...
1
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1answer
713 views

Understand Logarithm of Bar values manipulation step.

Currently I am learning Logarithm , but I can't understand the manipulation of the following Highlighted step how it comes How the result come after after ...
1
vote
3answers
116 views

How to prove that $\ln(1+x^2)<x$ , given that $x>0$

Given that $x>0$, how to prove that $\ln(1+x^2)<x$? I have been thinking about Taylor series, but didn't know how to do it. any suggestions?
6
votes
9answers
135 views

Find the value of $\log_8 9 \times \log_9 10 \times \cdots \times \log_n(n+1) \times \log_{n+1}8$

I'm completely lost on this question. I've been Googling around to no success. Find the value of $$\log_8 9 \cdot \log_9 10 \dotsm \log_n(n+1) \cdot \log_{n+1}8$$ I'm completely stumped as to ...
1
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2answers
45 views

Domain of $\log\lvert x^3+1\rvert$

This is a really simple question I think, but I'm looking for justification/clarification as well. I have a function \begin{align} y\left(x\right)=\log\left|x^3+1\right|,\tag{1} \end{align} state this ...
4
votes
5answers
60 views

Why is $\lim_{x \to +\infty}\log x = +\infty$ if $\mathrm{d}/\mathrm{d}x (\log x) = 1/x$?

Why is $\lim_{x \to +\infty} \log(x) = +\infty$? I would have expected that the value of this limit is some fixed number, since $$\frac{\mathrm d}{\mathrm dx} \log x = \frac1x$$ and $$\lim\limits_{...
1
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1answer
50 views

Rewriting log of a sum

Suppose we have a vector $$X=[x_1,x_2,\ldots,x_n],\quad x_i\in \mathbb{R} \text{ for } i=1,2,\ldots,n $$ Now if we have a formula $$f_X(x)=\log\left(\sum\limits_{i=1}^nx_i\right)$$ Is it possible to ...
7
votes
1answer
256 views

ln(z) as antiderivative of 1/z

When integrating $$\frac{1}{x}$$ (where $x \in \mathbb{R} $) one gets $$ln|x|+c$$ since for $x>0$ $$(ln|x|+c)'=(ln(x)+c)'=\frac{1}{x}$$ and for $x<0$ $$(ln|x|+c)'=(ln(-x)+c)'=\frac{1}{-x}(-1)=\...
0
votes
2answers
62 views

Natural Logarithm can't understand properties

~I don't get some of the properties of natural logarithm ($\ln$). $\ln(x^y) = y\ln(x)$ ex. $3\ln 7 = \ln 343$ and what is the difference between the above example and this $3\ln^2(7)$ not equal to ...
1
vote
1answer
12 views

Explaining the non-application of the multiplication law of logarithms, when logs are in the denominators.

I have an A' Levels student who had to solve the following problem: $ log_2 x + log_4 x = 2$ This was to be solved using the Change of base rule, and then substitution, as follows: $ \frac{1}{log_x ...
0
votes
1answer
14 views

Equations transformations with roots

How does the following transformation works (do not write that it is easy i want the answer): $$\ln \sqrt[n]{\frac{n!}{n^n}}=\frac{\ln \frac{n!}{n^n}}{n}$$
2
votes
1answer
75 views

Understanding the proof behind $\pi(x) \ge \frac{\log 2}{2}\frac{x}{\log x}$

I am trying to understand the argument behind the proof that: Given: $$\pi(2n) \ge \log 2\frac{2n}{\log 2n}-1$$ Then for $x \ge 2$: $$\pi(x) \ge \frac{\log 2}{2}\frac{x}{\log x}$$ Here's the ...
30
votes
3answers
1k views

Closed form for $\int_0^\infty\arctan\Bigl(\frac{2\pi}{x-\ln\,x+\ln(\frac\pi2)}\Bigr)\frac{dx}{x+1}$

I'm trying to find a closed form for this integral: $$I=\int_0^\infty\arctan\left(\frac{2\pi}{x-\ln\,x+\ln\left(\frac\pi2\right)}\right)\frac{dx}{x+1}$$ Its approximate numeric value is $$I\approx3....
1
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0answers
32 views

what is $\sum_{n=1}^{\infty}\frac{x^{n+1}}{n}-\sum_{n=1}^{\infty}\frac{x^n}{n}$

I posted a question earlier about the taylor of $(1-x)\ln(1-x)$ but i made a miscalculation and decided to delete it, sorry about that. anyways, i solved the miscalculation and i found that $(1-x)\ln(...
0
votes
0answers
51 views

trouble understanding complex logarithms

I am finding the complex logarithm very hard to understand. My text defines $G = \mathbb{C} - \{z \in \mathbb{C} : \Re(z) \leq 0, \Im(z) = 0\}$ and defines the principal logarithm to be the branch of ...
1
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0answers
26 views

I'm interested in the solution set satisfying the equation $\log_{10} p\times\log_{10} q=\log_{10} r$

The equation interested in is $\log_{10} p\times\log_{10} q=\log_{10} r$ where $p,q,r\in\mathbb N$ are natural numbers. Here, I want not to consider some trivial solutions that make any one of ...
2
votes
0answers
42 views

Proof of an inequality about primes

I'm very new to number theory and looking for a proof of the following inequality: $$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c \log^{...
2
votes
1answer
47 views

Asymptotic solution to inequality $x < k \ln(1+x)$

What is an upper-bound on $x$, given that $x < k \ln(1+x)$? I believe that the solution is something of the form $\mathcal{O}(k \ln k)$ but I am unable to prove this. This is my first encounter ...
2
votes
3answers
69 views

Why does $\int\frac{1}{2x+1}dx=\frac{1}{2}\ln|2x+1|+C$?

The way I am thinking is as follows: $$\int\frac{1}{2x+1}\,dx = \int\frac{1}{2}\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\int\frac{1}{x+\frac{1}{2}}\,dx = \frac{1}{2}\ln\left|x+\frac{1}{2}\right|+C$$ ...
0
votes
1answer
242 views

Interesting problem in logarithms

I know this place isn't for math problems/homework, and believe me I've been trying for a long time to solve this problem (45 mins to 1 hour) and besides I think many would find this useful or at ...
0
votes
2answers
41 views

I'm not great with logarithms so I'd appreciate some help with the following

How is it that $n^{\frac{1}{\log n}} = 10$. I understand that $10^{\log a} = a$ but I don't know how to make the correct algebraic manipulations. Note: Assume $log$ is base 10
1
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1answer
150 views

Forward Algorithm Hidden Markov Model matrix help [Discrete]!

So this may seem like a bioinformatics question but it is the math part that is giving me trouble. I'm using a Python package called YAHMM to model DNA sequences. I created a model with two states (...
0
votes
2answers
33 views

Is a dollar gain truly equal to a dollar lost?

Under the context of the asymmetrical nature of gain to a loss, as shown below, is 1 dollar gained truly equal to 1 dollar lost? If not how would you go about calculating the equalization ratio at ...
2
votes
3answers
47 views

Integral of $\frac{1}{2x}$

The integral of $\frac{1}{2x}$ is $\frac{\ln(x)}{2}$, but can't it also be $\frac{\ln(2x)}{2}$ or $\frac{\ln(3x)}{2}$? Is there a special reason for $\ln(Ax)$ to have identical derivatives?
8
votes
6answers
206 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
2
votes
4answers
84 views

How to solve $4x-\log(x) = 0$

I have a problem solving this equation: $4x-\log(x) = 0$. I can't seem to get this equation to a simpler form featuring $\log$s only or getting rid of the $\log$. Is there a way to solve it without ...
0
votes
1answer
34 views

Best fit in logarithmic chart

I have several variances ($\sigma^2$) which value depends on the velocity ($v$). As you can see in the graph, if increase the velocity, the variance does the same. I am studying this dependency, but ...
2
votes
2answers
29 views

Evaluating Logarithmic Expressions

Evaluate: $$\log_4 \left(\dfrac{1}{256}\right)$$ I am not sure how to approach this since there is nothing set equal to it.
3
votes
1answer
245 views

Integration of (Tsiolkovsky) rocket equation

The (Tsiolkovsky) rocket equation states that the velocity of a rocket can be calculated as $$v(t) = v_0 \ln\frac{m_0}{m_0 - \dot m t}$$ where $m_0$ is the starting mass, $\dot m$ is the (constant) ...
0
votes
1answer
32 views

How to normalize data in another scale?

Let $A$ be a set of values $\{a_1,a_2,a_3,a_4,a_5\}$ where $a_1 = 2$, $a_2 = 1$, $a_3 = 4$, $a_4 = 1$ and $a_5 = 2$, so, the $avg(A) = 2$. I'm looking for a normalization where the values below the ...
2
votes
0answers
112 views

Finding how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$ and logarithmic inequality question

Find how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$. $f<0$ at $x\to \infty$ and $x\to 0$. The derivative has a solution only at $x=\frac 1 k$. So place that point in $f$ and we'...
0
votes
3answers
63 views

Simplifying Quadratic Equations In Logarithmic Form

$log_{10}(x^2-x-7)=0.1$ $log_{10}(x-8)=1-log_{10}(x+1)$ $log_{10}(x+9)=1+log_{10}(x+1)-log_{10}(x-2)$ Note: I solved them as follows: $x = 3, -2$ but the textbook i'm using said there was no ...
3
votes
1answer
31 views

Prove convergence of this generalized integral

Prove the convergence of $$\int_0^1 \left[\ln\left(1+\frac1x\right)\right]^a\mathrm dx$$ for $ a>0$.
0
votes
2answers
99 views

Compute $\lim\limits_{n\to \infty} \ln(3n+7) - \ln(n)$

The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit. I need to find $\lim_{n\to\infty} \ln(3n+7)-\ln(n)$. I noticed that as $n$ ...
1
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2answers
60 views

Solve the logarithmic equation by $x$

Solve the eqation for all real $x$: $\log_2(x^2+7)+\log_3(x+6)=6$. What I tried: $\log_2(x^2+7)=a$ and $\log_3(x+6)=b$, then $a+b=6$ and $2^a=3^{2b}-4\cdot3^{b+1}+43$. But the problem is $a$ and $b$ ...