Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,256
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Inequality for log-convex functions
Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex.
Are these ...
- 93
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How is the dilogarithm defined?
I am pretty happy with the definition of the "maximal analytic contiuation of the logarithm" as
$$
\operatorname{Log}_{\gamma}\left(z\right) =
\int_\gamma ...
- 424
3
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2
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Why does the scalar inside a natural log dissapear when differentiating it? [closed]
For example if I was differentiating $\ln(2x)$ doesn't the chain rule dictate that it should be $2/x$, not $1/x$? Why does the $2$ disappear?
user1329181
8
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Approximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference
$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$
It's ...
- 47.3k
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How does $\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$?
Perhaps a basic question, as I was told that this was easily solvable using the chain rule, although I can't figure out why:
$\frac1{f(x)}\cdot\frac d{dx}f(x) = \frac d{dx}[\ln f(x)]$
I am wondering ...
- 204
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3
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Prove that a logarithmic function has maximum $0$
How do I prove that $$\log_{1/3} (|x-3|+1)$$ has maximum value $0$?
Do I have to equate this log function and $0$ to find out if it equals $0$, or I need to solve this some other way?
(Feel free to ...
- 57
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Unable to solve a logarithmic equation
I'm trying to solve this logarithmic equation for a while now, but I'm not getting any concrete solution.
$\log_{3x+8}(x^2 + 8x + 16) + \log_{x+4}(3x^2 + 20x + 32) = 7$
I defined the domain, converted ...
3
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2
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logarithmic inequality $\log_x64 < 2$
I tried to solve this inequation and I got the solution.
So $\log_x2^6=6\log_x2$, then i divided by 6 both sides and I got $\log_x2<\frac{1}{3}$.
I multiplied by 3 and put it into logarithm and got ...
- 41
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Why roots aren't the inverse of exponentiation but logarithms?
I think it's easy to see it when we look at the inverse of the function "$f(x) = a^x$" but I wonder if there's other way to look at it besides just analyzing the function. I was taught my ...
- 21
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Proof of absolute value of complex logarithm
I would like to prove that for a complex number z: $$\left| \text{Log}(-z) \right| \le \log \frac{1+|z|+|1+z|}{1+|z|-|1+z|}$$
Through numerical examples I already found that the equality only holds ...
- 41
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2
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Is there a closed form for the integral $\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\mathrm{d}z$?
As the title says, I would like to know if there is a closed form for the integral:
\begin{align*}
\int_{0}^{\infty}\ln(z)z^{\lambda - 1}\exp\left(-\frac{w}{2}\left(z + \frac{1}{z}\right)\right)\...
- 31
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The same equation giving different integrals?
I feel like I’m missing something obvious. I have checked on online integral calculators and I keep getting different answers despite the fact they are equivalent fractions.
$$\frac{1}{0.5x+5}=\frac{2}...
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8
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1
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how to integrate $\int_0^1 \ln^4(1+x) \ln(1-x) \, dx$?
I'm trying to evaluate the integral $$\int_0^1 \ln^4(1+x) \ln(1-x) \, dx,$$ and I'd like some help with my approach and figuring out the remaining steps.
or is it possible to evaluate $$\int_0^1 \ln^n(...
-1
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4
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Calculate $\lim_{n\to \infty}\int_n^{n+1}\frac{1-\ln{x}}xdx$ [duplicate]
$$
\mbox{Given}\ \operatorname{F}\left(x\right) \equiv
\int_{1}^{x}\frac{1-\ln\left(t\right)}{t}\,{\rm d}t,\ \mbox{calculate}\ \lim_{n \to \infty}\left[\operatorname{F}\left(n + 1\right) - \...
- 345
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If $\log_7 5$ = a , $\log_5 3$ = b , $\log_3 2$ = c, then the logarithm of the number 70 to the base 225 is?
So, I've tried using the properties:
$$\log_a b = \frac{\log_c b}{\log_c a}$$
and..
$$\log_a bc = \log_a b + \log_a c$$
And, the final simplification should be in the following options:
$$A. \frac{1-a+...
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