Questions related to real and complex logarithms.

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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 ...
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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 ...
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3answers
46 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
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2answers
82 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
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3answers
408 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 ...
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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! ...
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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 ...
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2answers
37 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: $ ...
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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) - ...
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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
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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
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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.
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1answer
46 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 ...
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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.
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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+ ...
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0answers
24 views

Convert log percentage to linear percentage

I am being provided some numbers in computer code between 0 and 1 that represent a percentage of a base 10 log scale that I need converted to a linear percentage. For example the 50% number for the ...
0
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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
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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 ?
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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
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3answers
64 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 ...
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2answers
28 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) ...
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2answers
76 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
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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
62 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
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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 ...
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2answers
65 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 + ...
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2answers
45 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
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0answers
53 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
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2answers
36 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?
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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
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1answer
38 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
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0answers
24 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. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
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0answers
24 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 ...
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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.
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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? ...
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5answers
53 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 ...
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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 ...
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2answers
55 views

Why is $x^2\ln\sqrt{x}$ equal to $\frac{x^2}{2}\ln x$?

My textbook jumps from $$x^2\ln\sqrt{x}$$ to $$\frac{x^2}{2}\ln x$$ What intermediate steps occur?
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1answer
27 views

how to find the inside value of logarithm?

I m doing sums in chemistry of first order reaction. In it, 0.521 = log(0.3/C) Then how to find the value of c?? The value is c= 0.09
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2answers
19 views

Exponential decay involving logarithm [closed]

In 2011 reactor $X$ released $4.2$ times the amount of cesium-137 as was leaked during reactor $Y$ disaster in 1986? Using; A = Pert Half-life = $30.2$ years. a) What year will ...
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1answer
40 views

Finding the intersections between $y = e^x$ and $y = x + 2$ algebraically?

In trying to find the intersections between $y = e^x$ and $y = x + 2$ in terms of $x$, I came up with the equation, $e^x = x + 2$ and subsequently, $x = ln(x+2)$. Beyond that point, I am stumped. ...
2
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3answers
67 views

The growth rate of $(\ln(x))^n$ is a lot slower than I expected

Obviously, the growth rate of $(\ln(x))^a$ is less than the growth rate of $(\ln(x))^b$ as long as $a>b$. Also, the growth rate of $(\ln(x))^n$ is apparently always less than the growth rate of ...
1
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1answer
22 views

Use of asymptotically equivalent equations in limits

I was wondering about the steps to show that the following limit does not exists. $$\lim_{x\rightarrow\infty}[\log(x^2-3)-\log(x+2)]$$ I know that by using L'Hopital's Rule and the continuity of ...
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0answers
41 views

Removing exponent from equation

I'm trying to solve the following equation numerically: This is problematic, because the term $ t^{\beta_i}_{j} $ becomes extremely large ($> 10000^{300}$), and unrepresentable with typical ...
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1answer
15 views

Complexity $\text{O}\left(\log(\log n))^{10}\right)$ vs $\text{O}\left((\log(\log n))^5\right)$?

If the question is not clear, then assume $t=\log(\log n)$, then the question can be re-framed as $\text{O}(t^{10})$ vs $O(t^5)$? So which has a higher order of growth? Thanks.
2
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3answers
71 views

How to show that $n^{\ln(\ln(n))} = \ln(n)^{\ln(n)}$

I have verified that $n^{\ln(\ln(n))} = \ln(n)^{\ln(n)}$ by plugging in values for $n$, but do not understand why it is true. I am not aware of any $\log$ rules that can be used to simplify ...
3
votes
2answers
89 views

$n$-th derivative of $\log(1+x)/x$

What is the $n$-th derivative of $$\frac{\log(1+x)}{x}$$ Now I have seen some analytical methods of getting $n$-th derivative of nicer looking functions such as the $n$-th derivative of $$ 1\over ...
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1answer
30 views

Can someone help me out with this question about logs? please

$$3^{2x}-2^{2y}=17$$ Find $x+y$. Here is what I did so far: Let $m=3^{2x}$ and let $n=2^{2y}$ $x=\frac{\log_3m}{2}$ , $y=\frac{\log_2n}2$ $$x+y= \frac{\log_3m+\log_2n}{2} $$ ...
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0answers
37 views

About defining logarithm on complex plane

"Is it possible to define $\log(z-1)$ continuously on $\mathbb C \setminus [-1,1]$? How about $$\log\frac{z+1}{z-1}$$ on $\mathbb C \setminus [-1,1]$? Why/Why not?" How should I approach these ...
2
votes
4answers
62 views

Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $=$ $32$ and $\log_3(x+y)+\log_3(x-y)=1$

Question: Solve for $x$ if $4^{\frac{x}{y} + \frac{y}{x}}$ $= 32$ and $\log_3(x+y)+\log_3(x-y)=1$ My attempt: With the first equation $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...