Questions related to real and complex logarithms.

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1answer
59 views

Why would $\forall x\log(x) = 0 \implies 2^\frac{1}{n} - 1 \leq \frac{\epsilon}{n}$ for large $n$

Why would $\forall x\log(x) = 0 \implies 2^\frac{1}{n} - 1 \leq \frac{\epsilon}{n}$ for large $n$? I'm reading a calculus text which used this in a reductio to prove the log function is nontrivial and ...
4
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1answer
35 views

How is $ \sum_{n=1}^{\infty}\left(\psi(\alpha n)-\log(\alpha n)+\frac{1}{2\alpha n}\right)$ when $\alpha$ is great?

Let $\psi := \Gamma'/\Gamma$ denote the digamma function. Could you find, as $\alpha$ tends to $+\infty$, an equivalent term for the following series? $$ \sum_{n=1}^{\infty} \left( \psi (\alpha ...
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3answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
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2answers
52 views

Integrating 1/x

The standard definition of integrating $\frac{1}{x}$ is: $$ \int \frac{dx}{ax + b} = \frac {1}{a} \ln |ax + b| + K $$ Now, if I'm understanding the "constant factor rule", that is: $$ \int k ...
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2answers
39 views

Proof that $\log(a^b) = b\log a$ when $\log$ is defined by an integral

When $\log a$ is defined as $\displaystyle\int_1^a\frac{dx}x$, then how does one prove that $\log(a^b)=b\log a$? I will post an answer here that is identical to an answer I posted to another ...
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4answers
77 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
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2answers
67 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
8
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2answers
162 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
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3answers
92 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
4
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1answer
45 views

$e$ and natural logarithms

How would you solve $6xe^{2x}+3e^{2x}=0$ for $x$ I tried: $\ln(e^{2x})=\ln(1/6x+3)$ $2x=\ln(1)-\ln(6x+3)$ $2x=-\ln(6x+3)$ but then I am stuck there. What am I missing?
3
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0answers
68 views

Closed form $\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}}$ and $\sum_{n=2}^{\infty} \frac{n}{\ln^n{n}}$

Apologies if this has been asked before, but I was playing around with Wolfram Alpha and got approximations but not closed forms for $$\sum_{n=2}^{\infty} \frac{1}{\ln^n{n}} \approx 3.2426094109 $$ ...
3
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0answers
47 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
2
votes
1answer
65 views

What's an intuitive way to compute summation of this series?

What's an intuitive way to compute $$\log(1)+\log (2)+\log (3)+\cdots+\log (n-1)+\log (n)$$ or for $n>a$ $$\log(a)+\log (a+1)+\log (a+2)+\cdots+\log (n-1)+\log (n) $$ I know the answer for ...
4
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5answers
378 views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
2
votes
4answers
129 views

What is the integral of x/ln(x)?

Well, I'm french so excuse me if I make some mistakes in english... I have to calculate this integral : $$ \int_{e}^{2e} \frac{x}{\ln(x)} dx $$ But I don't know how, can you help me please? Thank ...
1
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3answers
37 views

Solve x in logarithm equation

I am trying to solve $x$ for $2log_{10} (x-4) - log_{10}4(x-1) = 0$ I have the key with the answer 10 and have confirmed this is correct using Wolfram Alpha but which steps should I take to reach ...
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2answers
231 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
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3answers
42 views

Trouble with Logarithmic Differentiation

Hey guys I'm trying to find the derivative of this equation using logarithmic differentiation but I'm having some trouble. Wolfram Alpha is giving me different answers and I'm having difficulty ...
3
votes
2answers
138 views

Exponential function to logarithmic function

i'm stuck on completing this equations. Is this correct? $$z=a e^{-bt}$$ $$\ln(z)=\ln(a)+\ln(e^{-bt})$$ $$\ln(z)=\ln(a)+(1)(-bt)$$ $$\ln(z)=\ln(a)-bt$$
2
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1answer
32 views

Number of integers satsifying inqualities with logarithm

I am trying to solve the problem of finding the integers x satisfying the inequalities: $2\lt log_x45\lt3$ I realize this is a very basic question on logarithms and I have the key with the answers 4, ...
1
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1answer
28 views

Order of convergence for the method of false position

I'm reading about the order of convergence of the method of false position and there is one tricky point in the proof I don't understand. The method itself for finding the minimum $x^*$ of a function ...
3
votes
3answers
99 views

Trouble evaluating the sum involving logarithm

I was trying to solve this problem: Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$ In the procedure I followed, I came across the following sum: ...
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1answer
70 views

Why does $\log_{4}32 \neq \log _{4}(4 \cdot 8)$

$$\log_{4}32=2.5$$ If $$\log_{a}(b\cdot c) = \log _{a}b + \log_{a}c \,\,\,; (a>0, b>0,c>0, a\neq 1)$$ Then why does $\log_{4}32$ can't be $\log _{4}(4 \cdot 8)= \log_{4}4+\log_{4}8 = ...
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3answers
44 views

Logarithmic equations with different bases

I had problems understanding how to solve $$ 6^{-\log_{6}^2} $$ Any help would be much appreciated. Thanks!
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0answers
25 views

Equivalence of criteria using logarithmic transformation

Is the following criterion: $$ \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x} $$ Equivalent to: $$ \frac{\partial^2 \ln f}{\partial x\partial y} = ...
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0answers
41 views

Little o notation inequalities involving $n^{\log n}$

Apologies as this is a minor re-post, but I didn't think the other would get answers as it diverged into a discussion and got pushed down... I'm struggling with asymptotic notation a little bit... ...
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2answers
138 views

Solving ln/exponent question

How do I change the subject of the equation from x to y in the following equation: $$x=[4.105-\ln(\sqrt{y})]^2$$
10
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2answers
263 views
+50

A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
0
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1answer
38 views

True or false logarithmic branches

Say whether the following are true or false. Give a short proof. 1) $log(-z)+i{\pi}$ is a branch of the logarithmic function whose branch cut is the non-negative real axis 2)If $g(z)$ is a branch of ...
0
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1answer
18 views

Is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true?

Assuming $\mathbf{x}\in \mathbb{R}^n$, $f(\mathbf{x})\gt0 \forall\mathbf{x}\in\mathbb{R}^n$, is $argmin_{\mathbf{x}} f(\mathbf{x})=argmin_{\mathbf{x}} \log{f(\mathbf{x})}$ always true? Why?
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0answers
52 views

Can't find solution to Calculus 8th (Adams, Essex) problem

I've been sitting here for hours trying to find a solution to his problem. If you have the function $g(y)$, which is the inverse of $f(x) = x^x,\\ e^{-1} \leq x < \infty,$ show that ...
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5answers
118 views

How I could show that :$\log1=0$?

I would be like somone to show me or give me a prove for this : Why $\ln 1=0$ ? Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$. Thanks ...
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0answers
12 views

Natural logarithm of a square matrix without eigen-analysis

I'm trying to find a method to determine the natural logarithm of a square nonsingular matrix without using eigenvalues or eigenvectors. So far, I've only found this method: ...
3
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0answers
117 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
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1answer
80 views

Irrational to power of itself is natural

I've been thinking about a natural number like $n$ so that $x^x=n$ for some irrational $x$ but i couldn't find anything. As i didn't know how to approach the problem at all, i tried to make some ...
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2answers
27 views

Value of the logarithmic expression [closed]

Why $\log_6 \sqrt{6} = 1/2$? I know that $\sqrt{6} = 6^{1/2}$.
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4answers
32 views

Find the value of the Logarithmic Expression

Why is $\log_6 1$ equal to $0$ ?
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3answers
31 views

Solving for the value inside a base 10 logarithm

I have an equation of $\log(d)=(-x-A)/(10n)$ that I need to solve for $d$. How do I "reverse" the logarithm to obtain $d$? I apologize if this is super easy, I just can't even figure out how to Google ...
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0answers
26 views

Prove natural log between two finite harmonic sums [duplicate]

Prove for n in the naturals we have: $$\sum_{k=2}^n 1/k \le \ln(n) \le \sum_{k=1}^{n-1} 1/k$$ Intuitively this makes sense to me but I can't for the life of me figure out how to start this proof.
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0answers
29 views

Calculating log and trigonometric functions using only +,-,/,*

How to calculate logarithm and trigonometric functions (sin, cos etc.) on base n with using +,-,/,* ? Is there any way to do it?
2
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5answers
438 views

I have issue in calculating log values

How is $$log_42= \frac{1}{2}$$ ? Any formula to how we calculate this? I know i am confused when base is larger digit than log value term.
1
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2answers
66 views

Can someone suggest a function to achieve this graph?

I need a function that achieves a curved, logarithmic (I think) graph. The graph plots the level progression of a player of a simple game over time. Some players might progress faster, others slower, ...
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0answers
41 views

solving the logaritham [duplicate]

I was trying to solve: $\log_2 x \cdot \log_{0.25} x \cdot \log_{0.125} x \cdot \log_{16} x > \frac {2}3$ heres my attempt at it; using logaritham laws and a little algebra we get from $\log_2 x ...
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1answer
25 views

solving equation (indices/logarithms)

I don't really understand the Logarithms concept when it comes to question with ln or log with base e.for example question like this: 1.solve the equation $$x^4\mathrm{e}^{-2\ln(x)}=18-3x$$
2
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1answer
97 views

Calculate $I= \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}dx$

Please help me solve this: (level = high school) $$ \int_{1}^{e}\frac{(1+\ln x)x}{(1+x\ln x)^2}\,dx $$ Thanks
0
votes
1answer
23 views

What is the minimum degree of x so that it is greater than or equal to ln(x)?

I was thinking of this question and couldn't find it anywhere. I was trying to find a solution by finding the maximum of the function n = ln(ln(x))/ln(x) but I'm not sure if that's gonna work. Thanks ...
1
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1answer
35 views

Why does the same inequality give different answers?

$\left(\log _2\left(x\right)-2\right)\left(\log _2\left(x\right)+1\right)<0$ has a solution $\frac{1}{2}<x<4$ But when we take the second part alone that is $\left(\log ...
0
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1answer
38 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...
1
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3answers
55 views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
0
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3answers
88 views

How can I solve the following equation? [closed]

$\log_2{\frac{x-3}{x+2}}≤0$ Thank you.