Questions tagged [logarithms]
Questions related to real and complex logarithms.
1,000
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How to prove that $\log(x)<x$ when $x>1$? [duplicate]
It's very basic but I'm having trouble to find a way to prove this inequality
$\log(x)<x$
when $x>1$
($\log(x)$ is the natural logarithm)
I can think about the two graphs but I can't find ...
- 2,517
66
votes
12
answers
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Sum of the alternating harmonic series $\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{k} = \frac{1}{1} - \frac{1}{2} + \cdots $
I know that the harmonic series $$\sum_{k=1}^{\infty}\frac{1}{k} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \cdots + \frac{1}{n} + \cdots \tag{I}$$ diverges,...
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46
votes
11
answers
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Computing the integral of $\log(\sin x)$
How to compute the following integral?
$$\int\log(\sin x)\,dx$$
Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find ...
- 505
68
votes
4
answers
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Intuition behind logarithm inequality: $1 - \frac1x \leq \log x \leq x-1$
One of fundamental inequalities on logarithm is:
$$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$
which you may prefer write in the form of
$$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
- 1,908
39
votes
7
answers
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Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
user55225
23
votes
2
answers
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For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?
Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output.
I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and $\...
- 1,433
33
votes
6
answers
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Infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$
Recently, I encountered a problem about infinite series.
So my question is how to know whether the infinite series $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$ is convergent?
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8
votes
2
answers
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Find $\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^x\right)}{\ln\left(1+2^x\right)}}$
Prove the following limit without using approximations and derivatives: $$\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^{x}\right)}{\ln\left(1+2^{x}\right)}}=0$$
I cannot think of any possible ...
user171110
27
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6
answers
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Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $
It's my first post here and I was wondering if someone could help me with evaluating the
definite integral
$$ \int_0^{\Large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $$
Thanks in ...
- 279
42
votes
6
answers
30k
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Understanding imaginary exponents
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to boil down to ...
- 2,713
96
votes
11
answers
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Demystify integration of $\int \frac{1}{x} \mathrm dx$
I've learned in my analysis class, that
$$ \int \frac{1}{x} \mathrm dx = \ln(x). $$
I can live with that, and it's what I use when solving equations like that.
But how can I solve this, without ...
- 1,051
29
votes
7
answers
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Simple proof Euler–Mascheroni $\gamma$ constant
I'm searching for a really simple and beautiful proof that the sequence $(u_n)_{n \in \mathbb{N}} = \displaystyle\sum_{k=1}^n \frac{1}{k} - \log(n)$ converges.
At first I want to know if my answer is ...
- 1,173
9
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4
answers
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Show that, for all $n > 1: \frac{1}{n + 1} < \log(1 + \frac1n) < \frac1n.$ [duplicate]
I'm learning calculus, specifically derivatives and applications of MVT, and need help with the following exercice:
Show that, for all $n > 1$ $$\frac{1}{n + 1} < \log(1 + \frac1n) < \...
- 2,132
23
votes
5
answers
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Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
- 2,197
24
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15
answers
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Is there any simple method to calculate $\sqrt x$ without using logarithm
Suppose that we don't know logarithm, then how we would able to calculate $\sqrt x$, where $x$ is a real number? More generally, is there any algorithm to calculate $\sqrt [ n ]{ x } $ without using ...
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