Questions related to real and complex logarithms.

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0
votes
1answer
30 views

Numerical derivative of function wrt natural log of variable (non-analytic)

The function that I am trying to evaluate is $$ \frac{d y }{d \ln(x)} $$ where $d$ is the derivative. However I have a set of data points for $x$ and $y$ with uncertainties. Now I think that this ...
4
votes
1answer
67 views

Convergence/divergence of the sum $\sum_{n=2}^\infty 1/ \ln(n!) $

Is the sum $$ \sum_{n =2}^{\infty} \frac{1}{\ln n!} $$ convergent or divergent? I have tried different methods and it doesn't work. Perhaps comparing with a divergent series will work? I'm thinking ...
0
votes
1answer
41 views

Logarithms Problem, when finding $x^n = x$

Why is it that $1^4 = 1$, when using log laws why do you get $3 = \frac {\ln1}{\ln1} = 1 \therefore 2 = 0$? I was trying to show that $(-1)^{2^x + 1} = -1$, given $x \geq 0 $ and is an integer, but ...
1
vote
2answers
57 views

Why $n^{\log_2 n}$ almost equals $2^{\log^2_2 n}$?

Just by doing some calculations: ...
3
votes
1answer
37 views

Tight lower bound for logarithm function

Is there a lower bound for the logarithm function which is tighter than, $$\log(x)\geq 1-x^{-1}$$ that works for all real values of $x>0$?
-1
votes
1answer
17 views

Prove that Log is defined on D [closed]

$D=D(0,R)$ is the disk of center $0$ and radius $R$. Given that $a>R$ and $\Phi(z)=\frac{a-z}{a+z}$, I have proved that $\forall z\in D$, $\operatorname*{Re}(\Phi(z))>0$. Prove that $f = ...
0
votes
1answer
42 views

Show that $y=e^{e^{cx}}$ is a solution of the differential equation $\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$

Question: Show that $y=e^{e^{cx}}$ is a solution of the differential equation $$\frac{d^2y}{dx^2} =c^2 \cdot y \cdot \ln(y) (1+\ln(y))$$ I know there are a lot of ways of solving this ...
0
votes
4answers
96 views

Prove $\log(n!) =\Omega(n\log(n))$ [closed]

Can someone help me prove that $\log(n!) =\Omega(n\log(n))$, that is, that there exists some positive $c$ such that, for every $n$ large enough, $\log (n!)\geqslant c\cdot n\cdot \log(n)$?
0
votes
2answers
47 views

Rules of logarithm

Can anyone help me figure out how to go from the first expression to the second? $$ \begin{equation} \ln D=u+\delta(e-p)+\gamma y-\sigma r \end{equation} $$ $$ \begin{equation} \pi \ \ln (D/Y)= ...
0
votes
2answers
31 views

For what values are these logarithms true?

For what values, x and y, are both these equations true? $$\frac {\log(x)}{\log(y)} = \frac 23$$ AND $$\frac xy = \frac 23$$ How would one solve this?
1
vote
0answers
61 views

Natural Logarithm Integration

For what set of functions is $\int \frac{\ln{f(x)}}{f'(x)}\mathrm{d}x$ defined? More specifically related to the reason I ask, when $$ f(x) = \frac{c}{x} + \arcsin{x} + \sqrt{\frac{1}{x^2}-1} $$ is ...
0
votes
1answer
29 views

Solving an expression containing two added exponential functions

I have a problem solving the below equation with respect to $x$: $0.6\cdot \exp(\frac{-40}{x})+0.4 \cdot \exp(\frac{10}{x})=1$ My problem is that I have two exponential functions which are added ...
3
votes
2answers
37 views

logarithm equation with scalar on right hand side

$\log_7 (x^2-1) - \log_7 (x-1) = 2$ $\log_7 49 = 2$ => $\log_7 (x^2-1) - \log_7 (x-1) = \log_7 49$ => $\frac{(x^2-1)}{x-1} = 49$ => $x^2 -1 = 49(x -1)$ => $x^2 -1 = 49x -49$ => $x^2 - 49x + 48 = ...
2
votes
1answer
45 views

I am approximating $\ln x$ and $\log x$. How could I make these curves into a general equation?

Because I am waiting for my graphing calculator to ship, I need a quick-and-dirty way to calculate logarithms on a four-function calculator (for when I need to keep my laptop away from where I work). ...
1
vote
3answers
45 views

Derive $\log(a+b)=\log(a)-2\log\left(\cos\left(\arctan\left(\sqrt{\frac{b}{a}}\right)\right)\right)$

In the comments section of another post, MATHEMATIKER stated that $$\log(a+b)=\log(a)-2\log\left(\cos\left(\arctan\left(\sqrt{\frac{b}{a}}\right)\right)\right)$$ if $b>a>0$. I wish to know ...
1
vote
2answers
55 views

Find $b$ such that $\log_b(x)$ and $\log_b(y)$ are integers.

Is it possible to find a value $b$ such that, when given $x,y\in\mathbb{N}$, $\log_b(x)$ and $\log_b(y)$ result in integers? My assumption is that if $b\in\mathbb{Z}$, then $b$ may not exist, but ...
0
votes
3answers
47 views

how tell if a series of power numbers is bigger then others

I trying to order a list of mathematical expressions in string format as: "2*2" "4^1" "4^2^5" so far, so good for non exponential operations (^). I could compute ...
1
vote
2answers
84 views

Limit of the sequence $\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right]$

How can we evaluate the following limit $$ \lim_{n\to\infty}\frac{1}{n}\left[\log\left(\frac{n+1}{n}\right)+\log\left(\frac{n+2}{n}\right)+\dots+\log\left(\frac{n+n}{n}\right)\right] $$
10
votes
3answers
459 views

Number system with $e^x = 0$ for some $x$

It is well known that $e^x \ne 0$ for all $x \in \mathbb{R}$ as well as $x \in \mathbb{C}$. Upon reading this article and doing a bit of research I have found that this also applies to the ...
1
vote
1answer
73 views

How I can decompose $\ln(3f(x)+2g(y))$

I'm trying to simplify this equation: $$\ln(3f(x)+2g(y))$$ where $f$ is a function like $f=2x$ and $g$ is another function like $g=x²$ Can I rewrite this equation? Any help will be appreciated! ...
0
votes
4answers
72 views

Approximation $\log_2(x)$

Can anyone share an easy way to approximate $\log_2(x)$, given $x$ is between $0$ and 1? I'm trying to solve this using an old fashioned calculator (i.e. no logs) Thanks! EDIT: I realize that I ...
1
vote
2answers
38 views

How to find the unknown in this log inequality??

Find all values of the parameter a $\in\Bbb R$ for which the following inequality is valid for all x $\in\Bbb R$. $$ 1+\log_5(x^2+1)\ge \log_5(ax^2+4x+a) $$ I'm lost when I got to this stage: $ ...
3
votes
2answers
41 views

Finding the limit as $n \to \infty $ of $n\ln\left(1+\frac{\ x}{n^2}\right)$

Find $$\lim_{n\to \infty} n\ln\left(1+\frac{\ x}{n^2}\right)$$ My attempt: $\lim_{n\to \infty} n \left[\ln\left(\frac{\ n^2 +x}{n^2}\right)\right]$ = $\lim_{n\to \infty} n [\ln (n^2 +x) - ...
0
votes
1answer
20 views

Solve equation with variable in the fraction of a logarithm

I really had a hard morning thinking about how to solve an equation for a variable while the variable we want to solve for is in the fraction of a natural algorithm. I have this particular equation: ...
3
votes
1answer
69 views

Do those iterated increment rates always yield monotonic functions?

For any $a\in{\mathbb R}$ and any non-empty open interval $I$ containing $a$, we have an operator $T_a$ on $C^{\infty}(I,{\mathbb R})$ defined by $$ T_a(f)(x)=\left\lbrace \begin{array}{lcl} ...
1
vote
1answer
36 views

how to solve an nth derivative for the equation $\ln((1+x)/(1-x))$

I'm trying to find the $n$th derivative of this function. I've got that the first term is: $$ \frac{2(n!)x^{n-1}}{(x^2-1)^n} $$ Any improvement on this would be very helpful.
2
votes
1answer
48 views

What's wrong with my infinite series expansion for $\log(x)$?

Here, log is natural log. Looking at $f(x)=\frac{1}{x}$, I tried to put $f(x)$ in the form $\frac{a}{1-r}$ that an infinite geometric series $\sum_{n=0}^\infty (a \cdot r^n)$ converges to when $\mid ...
-7
votes
2answers
44 views

How to solve $\log(a^b)=b$? [closed]

How to solve for $b$ when log a common log: $$\log a^b=b$$ Please, denote the solution step-by-step. Any property denotation will also be very useful.
0
votes
1answer
34 views

How do I prove $\sum_{i=0}^{\log_3{n}}3^i = \frac{3n - 1}{2}$?

I started my data structures course at university and I came across with that equation, can someone explain me how I prove it please? $$\sum_{i=0}^{\log_3{n}}3^i = \frac{3n - 1}{2}$$ $$3^0+3^1+ ...
0
votes
1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
3
votes
1answer
96 views

integration of $\ln \ln x$

I would like to compute the following integral : $$\int_{2}^{\frac{\ln a}{\ln \ln a}} \ln \ln x \, \mathrm{d}x$$ where $a$ is a positive constant. Is this possible ?
0
votes
1answer
50 views

$\int_0^5 \frac{dx}{x^2-x-2}$

I am having some difficulty with this problem. I am getting a finite answer but when I put the equation into wolfram alpha to check my answer it says that the integral does not converge.Here is what I ...
0
votes
3answers
67 views

Approximating the value of a definite integral

I came across this question in ISI(Indian Statistical Institute) admission test $$I=\int_2^3 \frac{dx}{\ln(x)} $$ The four options were (A) is less than $2$ (B) is equal to $2$ (C) lies in the ...
0
votes
2answers
29 views

How to compute $=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$“by hand”?

The problem I'm having is with the logs. I go: $$\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}} \cdot \frac{n-2}{n-1} \Big)$$ $$=\lim_{n \to \infty} \Big( \frac{\log{(n+1)}}{\log{(n)}}\Big) ...
2
votes
2answers
77 views

Methods to integrate $(\ln x)^2 $ [closed]

What are some methods to evaluate the integral $$\int \left( \ln x \right)^{2} \, dx \hspace{3mm} ?$$
3
votes
1answer
31 views

How is the principal branch of logarithm defined?

In my textbook, it is defined as: $$\operatorname{Log} z = \ln |z| + i \operatorname{Arg} z$$ Where $\operatorname{Arg}$ is the principal branch of $\arg$, that's, the function which outputs the ...
2
votes
3answers
64 views

Intersection point of two functions - one linear, the other with logarithmic and sqrt terms

I would like first to appreciate everything that is being done on this forum and to greet you all! I have namely two functions and the goal is to find the intersection point of them. $y_1 = a + ...
2
votes
1answer
50 views

Stuck on this definite integral problem

I'm stuck on this definite integral problem. I need some constructive hint to proceed further. $$\int_0^a (a^2 + x^2)^\frac{5}{2} dx$$ Substituting $$x = a \cot\theta,$$ I have converted this ...
1
vote
2answers
67 views

Stuck on definite integral problem due to inappropriate $\log$

I have this definite integral problem which I have solved correctly but I'm stuck in one of the steps. I have manipulated it but I think it's not feasible to solve it that way. $$\int_0^a(a^2 + ...
0
votes
2answers
46 views

Sequence solutions of $ax=e^x$

This question comes from my answer to: Solving $4x = e^x$ without graphing and looking for intersection Here I've used a sequence of nested exponentials constructed from $$ x=\frac{1}{a}e^x $$ and a ...
0
votes
0answers
55 views

Why is numerical integration not working well on logarithm function with bounds $[-1,1]$

When I try to integrate function $x(\log(x)-1)$ from $-1$ to $1$, analytically I get $0.0000 - 1.5708i$ When I try to integrate it numerically, using $10$ points gaussian quadrature I get $0.0000 - ...
0
votes
2answers
37 views

Growth of debt: exponential, logarithmic, or linear? [closed]

If I have increasing debt that I don't intent to pay off for a really long time, how would I prefer to have it grow? Exponentially, logarithmically, or linearly?
0
votes
3answers
43 views

How to take log on this expression

I am solving exact differential equation, but I am stuck on the step on how to simplify this term or how to rewrite it. $e^{-2\ln{\sin{x}}}$
3
votes
1answer
40 views

Why is the discrete log problem intractable?

I have read the other questions on SE on this subject and they were not helpful to me, partially because I am not familiar with advanced mathematical notation. Let me explain the way I'm thinking of ...
2
votes
0answers
25 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
0
votes
0answers
27 views

logarithmic inequality with different bases and root

I have a problem with solving logarithmic inequality $$\log _{\frac{1}{5}}\left(\sqrt{x^3+x^2+x-14}\right)\cdot \log _{\frac{1}{4}}\left(-x^2+5x-6\right)<0$$ My attempt: The domain is ...
0
votes
2answers
33 views

If $x > y$, can you prove $x \log y > y \log x$, $x \ge 1$ and $y \ge 1$

If $x > y$, can you prove $x \ \log y > y \log x$, where $x \ge 1$ and $y \ge 1$. I encountered this problem in a paper I read and somehow cannot prove it.
0
votes
1answer
18 views

Population decline.

I'm looking at a question here and I'm a bit confused on how I'm supposed to solve it. A population of 460 decreases at 5% monthly. How many years will it take for there to be 100 left on the island? ...
2
votes
4answers
57 views

Solving a three-part log equation using the log laws

The question asks: Solve $$\log_5(x-1) + \log_5(x-2) - \log_5(x+6)= 0 $$ I know that according to log laws, addition with the same base is equal to multiplication and subtraction is equal to ...
0
votes
2answers
26 views

Getting rid off the logarithms in an equation to simplify

ok, I'm having trouble solving for equations when logarithms are involved. I know a little bit about logarithm rules but in equations I'm lost. example: $$\frac{1}{b}\ln{y}=\frac{1}{a}\ln{x}+c$$ I ...