Questions related to real and complex logarithms.

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1answer
58 views

Is this number in $O(\log(n))$?

Is this number $\big[\log(n) + \sum_{j=1}^{n-1} (\log(j) - (j+1)(\log(j+1)) + j \log(j) +1)\big] \in O(\log(n))$? I simplified it to $\big[\log(n) + \sum_{j=1}^n (-\log(n+1) - j(\log(n)) + 1)\big]$.
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2answers
51 views

$x^y = \exp( \ln(x) \cdot y )$, not a real solution for decimal numbers?

I am trying to understand how to calculate $x^y$ where $y$ is a decimal number, ($2^{2.3}$) According to wikipedia, the 'solution' would be $$ x^y = \exp( \ln(x) \cdot y ).$$ But if we break it ...
-1
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2answers
34 views

Approximate a summation

Approximate $3+ \displaystyle \sum_{x = 2}^{999}\dfrac{3(1000-x)}{1000+x}$. It may help to know that $\ln 2 = 0.69$. I was thinking of doing the integral test to approximate this but I am unsure if ...
0
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2answers
105 views

showing that the partial sums of $ \log(j) = n\log(n) - n + \text{O}(\log(n))$

I'm trying to show that the partial sums of $\log(j) = n\log(n) - n + \text{O}(\log(n))$ I know that $$\int_1^n\log(x)dx = n\log(n) - n + 1$$ so that this number is pretty close to what I want. Now ...
0
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1answer
73 views

$\log(xe^{2\pi i})=$?

I am confused about the differences of the properties of natural log in complex analysis and in real region. This question might be a bit stupid, but any answers or explanations of the log properties ...
8
votes
2answers
683 views

Roots of ln of a square

Problem: $$ y=\ln((3x-2)^2) $$ State the domain and the coordinates of the point where the curve crosses the x-axis At first sight, you say that the domain is $x>\frac23$ because $\ln$ is ...
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3answers
52 views

Why does $b^{\log_bx} = x$?

Why does $b^{\log_bx} = x$? Can someone break this down by showing me the steps as to why this is true?
1
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0answers
25 views

Converting log-scaled volume density to number fraction

I have a log scaled volume density distribution, $q_{3,log}$ from which I want to get number fraction, $\Delta Q_0$ with normal scale. So to transform $q_{3,log}$ to $\Delta Q_3$ the used relation is ...
0
votes
1answer
20 views

Find the dimensions of the rectangle with max area, base on positive x-axis, a side on the y-axis, and a vertex on y = e^(−x^2)

I know that this must maximize the definite integral from 0 to the x value, which would use the derivative of the integral but I'm unsure of how to set up the equation.
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0answers
40 views

logarithmic singularities in contour integration

How to evaluate the contour integral using the residue theorem if there is a logarithmic derivative? For example this: $$\int_C \log\zeta(s)\frac{x^s}{s} ds$$ or even this: $$ \int_C \frac{\log ...
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3answers
62 views

How do I prove that $f(x) = ln(x) − (x − 4)^2$ has exactly two roots.

I assume we take the derivative of the function. I get: $y' = 1/x-2(x-4)$ and I attempt to set it to 0 and solve but get stuck. Any tips?
0
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4answers
50 views

How do you find $y'$ for $x^y = y^x$?

Using the laws of logarithms: $y\ln(x) = x\ln(y)$, $y = x\frac{\ln(y)}{\ln(x)}$ Is it now quotient rule for the derivative? How is this done?
0
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4answers
46 views

find $y'$ for $y=(4+x^2)^x$

This differentiation requires the use of natural logarithms (the laws of logarithms), differentiation of logarithms, exponential function differentiation and the power rule. the formula for ...
1
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1answer
28 views

Solution of $\frac{c - 1}{x - c} + \log \frac{c - x}{x - 1} = 0$ for $x$

I was looking for the maximum of the function $f(x) = \left(x - 1 \right) \log\frac{c - x}{x - 1}$ for $\{x,c\} \in\mathbb{R}^+$, $x\not=1$ (obviously) and $x \le c-1$. The normal way to find such ...
2
votes
1answer
33 views

When does $\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$?

When does $$\log(\lim_{x\to c} f(x)) = \lim_{x\to c} \log(f(x))$$ I have seen different things from different sources. For example, this and this. Does $f(x)$ have to be continuous at c, does $\log ...
3
votes
3answers
70 views

Solve the equation,$\sqrt{\log(-x)}=\log{\sqrt{x^2}}$

Solve the equation,$\sqrt{\log(-x)}=\log{\sqrt{x^2}}$(base of log is 10) $\sqrt{\log(-x)}=\log{\sqrt{x^2}}$ $\sqrt{\log(-x)}=\log{|x|}$ Now two cases arise,when $x>0$ and when $x<0$ When ...
2
votes
2answers
33 views

$c$ is the value of $x^3+3x-14$ where $x=\sqrt[3]{7+5\sqrt2}-\frac{1}{\sqrt[3]{7+5\sqrt2}}$.Find the value of $a+b+c$

$a=\sqrt{57+40\sqrt2}-\sqrt{57-40\sqrt2}$ and $b=\sqrt{25^{\frac{1}{\log_85}}+49^{\frac{1}{\log_67}}}$ and $c$ is the value of $x^3+3x-14$ where ...
0
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1answer
27 views

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$.

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$. Let $x$ be a positive integer. Now the characteristic of ...
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0answers
30 views

Solving the curve equation for logarithmic decay using two anchor points.

I would like to have an adaptable logarithmic curve equation that I can then find y for any value of x. I have two points (x1,y1) and (x2,y2). My data requires constant decay (financial discounting ...
0
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3answers
21 views

How do I calculate the following logarithm?

Say I'd like to calculate the following logarithm: $$log_{0,1}{\sqrt {10}\over 100}$$ Using the logarithm rules, I do it this way: $${log_{1\over 10} {\sqrt {10}}} - {log_{1\over 10} {100}}$$ ...
0
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2answers
38 views

Complex but Simple equation.

Im( e^z ) = 1 I've reached rsin(t) = 1 but I am stuck here, where r is radius and t is argument. There may be use of logs here I might be missing.
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1answer
18 views

How can I solve this complex Log equation?

|Log z| = Re (Log z) z is complex Log z denotes the principle value of log z. -pi< Arg z< pi. Step by step solution please. I'll need to sketch it as well.
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2answers
48 views

Taylor series of $\ln(x+2)$

I try to determine the Taylor series of $\ln(x+2)$. Since I know $\ln(1-x) = \sum_{n=1}^{\infty} \frac{x^n}{n}$, I suppose I can rewrite, \begin{align} \ln(x+2) &= ...
2
votes
2answers
86 views

What is the value of $\ln(\ln(i))$?

I came across this question while practicing some quant interview questions. It simply asks the value of the above expression, no additional information is given. I tried googling for it, Google ...
2
votes
1answer
63 views

Why is $\ln(\sqrt{|2x-5|}) + \frac{1}{2} \ln(|2x+3|) \neq \ln(\sqrt{|2x-5|}) + \ln(\sqrt{|2x+3|})$ in Wolfram Alpha?

According to Wolfram Alpha, $$\frac{1}{2} \ln(|2x+3|) = \ln(\sqrt{|2x+3|})$$ is always true, which makes sense given what I know of log rules. However, if I add the expression $\ln(\sqrt{|2x-5|})$ ...
2
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1answer
34 views

$\lfloor\log{p_{n}}\rfloor$ having more than one solution for individual $k$

Question: If you assume that a. $k\in\Bbb{N}$ b. $p_n$ denotes the $n$'th prime number. $p_0$ doesn't exist. c. $n\in\Bbb{N}$ I am fairly certain that: At least two distinct integer values for ...
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2answers
43 views

An interesting log problem $3^{{(log_3{x})}^2}+x^{log_3x}=162$

$$3^{{(\log_3{x})}^2}+x^{\log_3x}=162$$ How do I go about doing this. I am stuck at the step $x^{\log_3x} = 81$. Is this right? How do I continue or is it wrong?
2
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0answers
48 views

Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove ...
0
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1answer
14 views

Question about the “method of increments” derivation of the derivative of $\log(x)$

From Morris Kline's "Calculus: An Intuitive and Physical Approach": At the end of the derivation (which I understand up to this point), he gets to $$\frac{\Delta y}{\Delta x} = \frac{1}{x_0} \cdot ...
2
votes
2answers
76 views

Error on Matlab and Wolfram Alpha?

I solved the differential equation $\frac{\mathrm{d} y}{\mathrm{d} x}= \frac{y^2-1}{2y}$, in the following way: $\frac{\mathrm{d} y}{\mathrm{d} x}= \frac{y^2-1}{2y} <=>$ $\int ...
2
votes
1answer
19 views

is my answer correct? derivative of logarithmic functions

I want to check my answer, pleas tell me if it's correct or not first problem $y=\left(\log _{\frac{1}{x}}\left(e\right)\right)$ my answer $y=\frac{lne}{ln_{\frac{1}{x}}}$ ...
0
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2answers
67 views

Complex logarithm: $\log(e^{5i})$

I am try to calculate: $\log(e^{5i})$, but I think I am doing something bad, I suppose that $e^{5i}=\cos(5)+i\sin(5)$, angle is $\tan(5)$ then... $\log(e^{5i})=\log(1)+i(\tan(5))$?, help. And more, ...
1
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2answers
53 views

Natural logarithmic derivative trick

Hi chaps and chapesses, I was wondering if someone could just explain something. If I have a function which is dependent on $x$, the familiar $f(x)$. Now, if I take the derivative of this, and ...
2
votes
1answer
102 views

Proof of an inequality including logarithms

$0 <x,y \le 1$ $|y \log y-x\log x|\le|x-y|^{(1-\frac{1}{e})}$ I want to know how to solve this inequality.
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2answers
29 views

Quick Log Rule Confusion

I want to simplify $$\log_{c}{-c}$$ I know that $$\log_{c}{c^k} = k$$ But is there a $\log$ rule for the first one?
3
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2answers
110 views

Riemann Zeta Function integral

I was reading about the Riemann Zeta Function when they mentioned the contour integral $$\int_{+\infty}^{+\infty}\frac{(-x)^{s-1}}{e^x - 1} dx$$ where the path of integration "begins at $+\infty$, ...
0
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0answers
51 views

Calculation of infinite product

My question is to prove the identity: $$ \prod_{n=1}^{\infty}\left(\frac{\cos t-1}{n}+1\right)=\exp\left(-\int_0^1x^{-1}(1-\cos xt)dx\right) $$ which arises as a product of characteristic functions of ...
2
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1answer
93 views

Table Maker's Dilemma

I am trying to understand the TMD based on this doc. As per my understanding, For transcendental functions i..e (log2 log10 1oge, exp,sin, cos,tan...) the exact value for an input cannot be ...
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0answers
24 views

How does the Naive Bayes Classifier become a sum of logs?

The Wikipedia article for Naive Bayes Classifiers explains how we come up with the equation However, it proceeds to simplify it to become I don't understand the logical process taken to reach ...
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3answers
123 views

Find $x$ in $\frac{(3x^2-27)(8x^2)^6}{4(9-3x)(x^2+3x)}=\frac{\tan(x+4)}{\log(x+\frac{1}{4})}$

Found this little puzzle on facebook, not sure if it was a joke. Find $x$ $$\frac{(3x^2-27)(8x^2)^6}{4(9-3x)(x^2+3x)}=\frac{\tan(x+4)}{\log(x+\frac{1}{4})}$$ I'm thinking LHS numerator: ...
0
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2answers
66 views

How to solve equations like $8n^{2} = 64n\log_2(n)$

How do I solve this equation: $$ 8n^{2} = 64n\log_2(n) $$ I've plotted logarithmic curves for both functions and found that they intersect at $n = 44$, but I've no idea how to solve the equation.
2
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1answer
28 views

Creating simple function to rate potential quality of a basketball game

The concept is relatively simple, but unsure how to implement it. I want an output of a number between 0 and 1 where the inputs are the sum of the ranks of the teams and the point spread. E.g. ...
1
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2answers
72 views

log of summation expression

I am curious about simplifying the following expression: $\log \left(\sum_\limits{i=0}^{n}x_i \right)$ Is there any rule to simplify a summation inside the log?
1
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1answer
31 views

Indefinite integral: chance of analytic solution?

Is there a chance the following indefinte integral has an analytic solution? $I(a, b) = \int \mathrm{d}x \sqrt{a^x +b^x}$ My attempts and those of Mathematica proved fruitless thus far... Any ...
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1answer
19 views

Need to know if the given du value for this integral/ln solution is correct

$\int \frac{2}{x(\ln x+5)^7} dx$ for that problem the example gives $u=\ln x+5$ and $du=\frac{1}{x}dx$. based on a couple other problems I thought du should be $\frac{1}{x+5}dx$ The derivative of ...
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1answer
25 views

Asymptotic complexity of power of logs

I'm trying to simplify $\Theta(lg^k(n/2))$. I believe it's $\Theta(lg^kn)$ but i don't know if the following proof is correct... i'd love to receive some input I tried doing - ...
1
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3answers
42 views

Solving logarithmic equation $(e^x+e^{-x})/(e^x-e^{-x}) = 3$ algebraically

I have the equation $$(e^x+e^{-x})/(e^x-e^{-x}) = 3$$ I can solve it using a graph, but how would I go about solving it algebraically.
3
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1answer
66 views

Prove independence of a pairwise independent subsequence of independent events

Consider infinite independent coin tossing where $H_n = \{$nth coin is heads$\}$ for $n = 1, 2, ...$. Let $$A_n = \bigcap_{i=1}^{\left \lfloor \log_2 n \right \rfloor} H_{n+i}$$ How do you show that ...
1
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2answers
47 views

Finding values of x for logarithm

The question is to find the numbers of x which satisfy the equation. $$ \log_x10=\log_4100. $$ I have \begin{align*} \frac{\ln10}{\ln x} &= \frac{\ln 100}{\ln 4} \\ \frac{\ln10}{\ln x} &= ...
1
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5answers
70 views

Proof of reciprocal logarithm

I need to prove this logarithm. $$\log_p\Big(\frac{1}{x}\Big) = -\log_p(x)$$ The first step would be $\ln(1/x)/\ln_p$ I need help as to what the next step would be.