All Questions
Tagged with logarithms real-analysis
478
questions
3
votes
1
answer
195
views
Generalizing a logarithmic inequality
Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$
$$\alpha_i = x_i (1-x_i) \log^2 \frac{x_i (1-y_i)}{y_i (1-x_i)}$$
and
$$\beta_i = x_i \log \frac{...
- 400
8
votes
1
answer
276
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Approximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference
$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$
It's ...
- 47.3k
1
vote
3
answers
89
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Prove that a logarithmic function has maximum $0$
How do I prove that $$\log_{1/3} (|x-3|+1)$$ has maximum value $0$?
Do I have to equate this log function and $0$ to find out if it equals $0$, or I need to solve this some other way?
(Feel free to ...
- 57
1
vote
1
answer
59
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Convergence for a sequence using logarithmic properties?
Given the sequence $x_n=\frac{\log(3n+2)}{\log(n^2+2)}$, which has the limit $\frac{1}{2}$ when n goes to infinity, give a formal proof of the limit using the epsilon definiton.
$$| x_n - L |< \...
- 41
1
vote
1
answer
99
views
Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms?
Are there other solutions to the functional equation $f(x^t) = t f(x)$ besides logarithms? Here $x$ and $t$ are real variables with $x>0$.
I know that given the property of logarithms $\log(x^t) = ...
- 1,474
6
votes
0
answers
156
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Finding a closed form for $ \int_0^1 \frac1x \ln\left(\frac{\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x $
I want a closed form for the following integral
$$
\int_0^1 \frac1x\;\ln\left(\frac {\ln\left(\frac{1-x}{2}\right)}{\ln\left(\frac{x+1}{2}\right)}\right)\, \mathrm{d}x
$$
An integration by parts ...
- 203
1
vote
1
answer
44
views
Bounding the solution of a logarithmic equation
Given a small number $\varepsilon >0$ and a constant $1/3\le \alpha < 1 $, I am looking for the smallest possible number $x^*$ such that for all real $x\ge \max\{x^*,3\}$, we have
$$\frac{x}{(\...
3
votes
6
answers
360
views
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm ...
- 1,474
2
votes
2
answers
118
views
does Integrating both sides of an equation in dx will Invalidates the equality?
I'm struggling to grasp the justification behind integrating both sides of an equation. While I understand that operations can be applied to both sides, maintaining equality, it appears that this ...
- 43
0
votes
1
answer
42
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Does the domain of the real logarithm need to be "extended" when dealing with even exponents in the argument?
Consider $f(x)=\log(x^2)$. Clearly, the domain of $f$ is $D_f=\mathbb{R}-\{0\}$, since $x^2>0$ for any $x<0$. However, by the fundamental properties of logarithms:
$$
f(x)=\log(x^2)=2\log(x)\...
- 67
0
votes
1
answer
99
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Showing that the supremum of a given function is finite.
Consider arbitrary elements $n \in \mathbb N$ and $ \lambda \in \mathbb R$ such that $0 < \lambda < n$.
Problem. My goal is to prove that the supremum
$$ \sup_{r > 0} f(r) $$
is finite, where ...
3
votes
1
answer
156
views
Simplify $n$th derivative of $\frac{\log^3 x}{1-x}$
I need to simplify $n$th derivative of $$f(x)=\frac{\log^3 x}{1-x}$$ where $0<x<1$
I tried writing $f(x)=u_1u_2u_3u_4$ where $u_1=u_2=u_3=\log x$ and $u_4=\frac{1}{1-x}$.
Using Leibniz rule for ...
- 926
3
votes
0
answers
78
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What functions satisfy $f(ax) - f(a(x-1)) > f(b(x+1)) - f(bx)$ for all $a, b \in \mathbb{R}^+$ and $x \in \mathbb{Z}^+$.?
I am looking at a family of functions $f : [0, \infty) \rightarrow [-\infty, \infty)$ satisfying the following property:
$$f(bx) - f(b(x-1)) > f(a(x+1)) - f(ax) \quad \text{for all $a, b \in \...
- 31
0
votes
1
answer
56
views
Equality after exponential function swap
The equation $ 2^x = x^2 $ has two real obvious solutions $(x=2,4)$ and another root not so obvious $x\approx- 0.766665$;
Assuming existence and uniqueness, extending equality $f(x)= g(x)$ between ...
- 41.1k
3
votes
0
answers
102
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How to tell if an infinite series sum will be rational or irrational?
I plugged in the following series into a calculator: $$\sum_{n=1}^\infty \ln(1+\frac{1}{n^2})$$ and got a result of approximately $1.29686$. That's nice and all, but I want to know: is this result ...
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