Questions related to real and complex logarithms.

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3answers
63 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) ...
2
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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
61 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 ...
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vote
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
votes
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
35 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
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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
votes
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
votes
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. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
0
votes
0answers
23 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.
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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
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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vote
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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votes
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 ...
4
votes
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
votes
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
vote
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 ...
1
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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 ...
0
votes
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
votes
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 $$4^{\frac{x}{y} + \frac{y}{x}} = 32$$ ...
1
vote
0answers
25 views

Find the distribution of $ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt .6\right\}. $

The Statement of the Problem: Let $ \{ U_i \}$ be a set (sequence?) of iid random variables such that $U_i \sim \text{Uniform}(0,1)$, and define $$ N = \min \left\{k: \prod_{i = 1}^{k}U_i \lt ...
0
votes
4answers
57 views

help needed to compute derivative of $e^{x\sin x}$

How should I compute the derivative of $e^{x\sin x}$ ? I am a student of class 11, so can you explain me how to do this without high level mathematics ( I know first principles ) I know that ...
2
votes
2answers
114 views

Integral of $\sqrt{ {\rm ln}^2 4 \cdot 4^{2 x} + 1}$

I'm currently taking calculus, and have hit a problem that is causing me confusion. I have the answer to the problem, I just have no idea how to arrive at that answer. The problem is as follows: ...
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votes
1answer
53 views

How do I prove that L(2) < 1 and L(3) > 1 given when only given the integral..

We know that $$L(x) = \int_{1}^{x} \frac{dt}{t}, \quad x> 0$$ And we cannot use that $L(x) =\ln(x)$. How do we prove that $L(2) < 1$ and $L(3) > 1$ ?
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0answers
66 views

Simply connected domains and complex logarithms

While studying Complex Analysis from my professor's notes I came across the following theorem. A demain $D$ in the complex plane is simply connected if and only if any analytic function $f(z)$ on ...
0
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0answers
64 views

How can be $\log(\sqrt{1-x^2})$ equal to $\log(\sqrt{x^2-1})$

In a book, $\log(\sqrt{1-x^2})$ is given as the solution of an differential equation. I by properly managing where minus sigh "-" occurs and using a different anti-derivative computing approach can ...
0
votes
1answer
41 views

Bounding a sum of logarithms

Consider a function $f:(0,\infty)\rightarrow \mathbb{N}$ with argument $\epsilon$. Suppose $f$ is decreasing in $\epsilon$. Let $0<b<1$, $K>0$, $d \in \mathbb{N}$, $\delta>0$. Assume $$ ...
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vote
1answer
35 views

Two anti-derivatives of $\int \frac{du}{1-\frac{1}{2}u}$

WolframAlpha computes $\int \frac{du}{1-\frac{1}{2}u}$ to be $-2\cdot\log(u-2)$. Computing derivative of this result confirms its correctness. However, I manually obtained the result ...
2
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1answer
44 views

Is there a simple way for natural logs be calculated by hand?

Why are natural logs not calculated by hand often? Is it too difficult to get a accurate answer without a calculator?
4
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4answers
43 views

How to solve the differential equation $\frac {dP}{dt} = \frac {1}{2} (P-200)$ when we are given that $p=200$ when $t=0$?

The title pretty much describes the question. Posting this on behalf of a friend. He says he knows how to apply the integrating factors method, but can't figure out how to solve this. Here's my ...
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3answers
44 views

Determine whether a number is a power of 3

Why does $\frac{\log(n)}{\log(3)}$ being an integer determine whether $n$ is a power of three? While doing some programming exercises I came across this problem and the above formula was a proposed ...
2
votes
1answer
76 views

why $3 - \sin{x}$ is always positive?

Hello I am currently learning integration and after integrating the function $\int \frac{\cos{x}}{3-\sin{x}}$ I end up with $-\ln{|3-\sin{x}|} + c$. However in the textbook it is stated that, since ...
0
votes
0answers
29 views

Non zero continuous path $[0,1]\to \mathbb C$ has continuous logarithm

Let $\gamma:[0,1]\to \mathbb C$ be continuous, and not passing through $0$. How can we prove that, using complex analysis, there is a continuous $G:[0,1]\to \mathbb C$ so that $\gamma=e^G$ ? This can ...
1
vote
2answers
40 views

Monotonicity of the sequence $(a_n)=(\frac{\ln n}{n})$

The solution of this problem in my book is the following "For $n\neq 1~~$ $\frac{a_{n+1}}{a_n}=\frac{n \ln(n+1)}{(n+1)\ln n}<1$, so $(a_n)$ is monotone decreasing." How can we write this ...
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1answer
18 views

Using either x-roots or logarithms to simplify equation

I have the equation: $.5 = 1-(1-s^r)^b$ And want to solve for $s$. Would the correct solution involve (1) b-roots and r-roots or (2) logarithms? Here's approach (1): Here's approach (2): ...
0
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1answer
22 views

Properly using natural logs to solve for a variable

I'm attempting to take a logistic graph and create an equation using the logistic model of continuous growth. I have taken the equation and simplified it down to ...
0
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1answer
50 views

The same derivative for very different functions

Just a curiosity and request for a comment on this. $$ \frac{d}{dx}(-2 \log (u x-2))=-\frac{2 \left(x u'\right)}{u x-2} $$ $$ \frac{d}{dx}(-2 \log (2-u x))=-\frac{2 \left(x u'\right)}{u x-2} $$ ...
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votes
2answers
28 views

Finding the interval of which a multivariable function is defined

Find the interval in which $f(x,y,z)=z+ln(1-x^2-y^2)$ is defined So all that is need to to check for which values $ln(1-x^2-y^2)\geq0$ That mean $1-x^2-y^2\geq 1 \rightarrow x^2+y^2\geq 0$ But ...
1
vote
8answers
178 views

How to prove $\ln x<x$?

How can we prove that the inequality $$\ln~x<x$$ It is trivial in the cases $0<x\le 1$. I couldn't do anything for $x>1$.