Questions related to real and complex logarithms.

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0
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1answer
16 views

How to solve this Diophantine equation (involving natural logarithms)?

The equation is $r = \ln{a} + b \ln{c}$ where $r \in \mathbb{R}$ is fixed and $a,b,c \in \mathbb{N}$. In other words, for arbitrary real r, how can one say whether a solution (in form above) exists ...
1
vote
2answers
40 views

How can I isolate for the $z$ exponent?

Can anyone help me with this math equation? Solve for $z$ $$P = \frac{e^z}{1 + e^z}$$ $$P(1 + e^z) = e^z$$ $$P + Pe^z = e^z$$ $$P = e^z - Pe^z$$ I've got this far, am I at least on the ...
0
votes
1answer
35 views

Need help with logarithmic differentiation

I need to use logarithmic differentiation to get f(x)=x$\sqrt{(x+1)(x+2)(x+3)(x+4)}$. I've been working on it for a while and could use some help. Thanks!
3
votes
2answers
67 views

Compute Power Series Convergence to a function

Consider the next power series $$ \sum_{n=1}^{\infty} \ln (n) z^n $$ Find the convergence radius and a the function $f$ to which the series converges. I have easily found that $R=1$ is the ...
2
votes
3answers
53 views

Prove that $\log_52$ is irrational

Prove that $\log_5(2) \in \mathbb{R}\setminus \mathbb{Q}$ (irrational numbers). I know there is a question out there already for this but my problem is that I need to prove this using the ...
0
votes
2answers
77 views

Solve $2^x=13 \bmod 3^4$

Solve $2^x=13\bmod 3^4$ I know $\log13=30\bmod 3^4$ and $\log16=15 \bmod 3^4 $ I've tried subbing $\log13/\log16$ for $2$ but I am not sure what to do next.
2
votes
1answer
82 views

How to solve this logarithmic inequality?

I've started a data structure course and I need some help with solving these logarithmic inequalities. It would also be helpful because later on these kind of calculation won't pose a problem later ...
1
vote
1answer
45 views

Could someone explain steps?

I am learining about logarithm equations, and i can´t seem to understand how to solve such an equation, could someone help? I must solve the equation/find $x$ for: $$2^{2x} - 3\cdot2^x - 10=0$$ The ...
1
vote
2answers
74 views

Find $\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$

How to calculate this limit? $$\lim\limits_{n \to \infty} \frac{\log(1+2^n)}{\log(1+3^n)}$$
1
vote
3answers
40 views

Question regarding logarithms 2

What is $\ln(-1)$? And would there a taylor series for $$\ln\frac{1+x^m}{1-x^m}$$?
0
votes
1answer
16 views

Solve for r. Logarithms

$$ 36000 = 3450 * \frac{1-[1/(1+r)^{12}]}{r} $$ The next step is divide both sides by 3450. Now I'm stuck. Help solve for r.
0
votes
2answers
134 views

Integration of 1/x as a limit of a sum

This is from R.Courant book Example "Introduction to Calculus and Analysis vol.1 " To integrate $x^\alpha$ when $\alpha\neq1$ we subdivide the interval [a,b] by the point of geometric progression: ...
0
votes
3answers
39 views

Question regarding logarithms

Can you factor out the $m$ out of $\ln(c\cdot x^m)$ where $c$ is a constant?
0
votes
1answer
49 views

Logarithm inequality for specific range

I need to show that: $$ \ln(1+x)\left(\ln\left(\frac{1+x}{1-x}\right)+1\right)+\ln(1-x)\ge 0, $$ for $0\le x\le 2/3$. Thanks
1
vote
0answers
14 views

Calculating gain ratio from a dB value

In a practice problem I have: power gain = $\log_{10}(\frac{db}{20})$ The final answer for the ratio is 1. The dB value is $-3$. When I do $\log_{10}(\frac{3}{20})$ I get $-0.823$. Just wondering ...
-1
votes
1answer
42 views

Why is $\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$ [closed]

Why is $$\frac{\sum_{i=1}^n \log(X_i)}{n} = \overline{log X}$$ ($X_i$ are i.i.d samples)
1
vote
2answers
98 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
votes
1answer
47 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 ...
3
votes
1answer
285 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
92 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 ...
-2
votes
4answers
66 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
47 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
vote
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
67 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
48 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
71 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
105 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
36 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 ...
1
vote
3answers
39 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
30 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 ...
1
vote
3answers
18 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 $ ...
0
votes
0answers
38 views

Order of operations for log transformation

I am working with a large dataset of positive values with a positive skew. I will be using a Ln transformation in SPSS to normalize my dataset. However, I am not sure of the order of operations. For ...
1
vote
1answer
114 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 ...
0
votes
0answers
30 views

on the sum of ordered log functions

I have the following question and I need some suggestions on how to address it. Assume we have the following non-increasing ordered positive constants (not variables) $a_i, i = 1, ..., N,$ (i.e., we ...
6
votes
9answers
119 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
vote
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
56 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 ...
1
vote
1answer
26 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 ...
0
votes
0answers
22 views

What are binary operation properties, so sequence fold could be mapped?

I have a sequence of numbers ${x_1, x_2, ..., x_n}$. The task is to compute a value of $S = f(x_1, f(x_2, ..., f(x_{n-1}, x_n)))$ where $f$ is one of: $f(x,y) = log(x \cdot y)$ $f(x,y) = x \cdot ...
7
votes
1answer
98 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 $x<0$ ...
0
votes
2answers
60 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
11 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
55 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 ...
28
votes
3answers
929 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 ...
1
vote
0answers
29 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 ...
0
votes
0answers
31 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
vote
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 ...
2
votes
1answer
37 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 ...