Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,256
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nth Root of a Rational Function
Suppose I have two polynomials $p(z)$ and $q(z)$ and a positive integer $n$. Suppose I wanted to define $r(z)=(\frac{p(z)}{q(z)})^{1/n}$ on $\Omega$ such that r(z) was analytic and single valued. On ...
- 1,454
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2
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Showing $\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right) \, dx=\frac{\pi^2 + 4}{2}$
While exploring possible applications for exponential substitution, I stumbled upon the following integral identity:
$$\int_{-1}^{1}\ln \left( \frac{x+1}{x-1} \right) \left( x - \sqrt{x^2 - 1} \right)...
- 1,011
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Unexpected examples of natural logarithm
Quite often, mathematics students become surprised by the fact that for a mathematician, the term “logarithm” and the expression $\log$ nearly always mean natural logarithm instead of the common ...
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Prove that sequence $(a_n)_{n\geq 1}$ is not convergent.
The standard branch of logarithm, $\log:\mathbb{C}\setminus (-\infty,0]\to\mathbb{C}$ is defined as
$$
\log(z):=\ln|z|+i\operatorname{Arg}(z)\tag{2}
$$
where $\arg(z)$ is the standard branch of the ...
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What is the product of the solutions to the equation (√10)(x^(log (x))) = x^2? [closed]
I have spent a few hours on this question but I can't seem to grasp it. I have found that taking the log of both sides doesn't give me any progress. The log exponent identity didn't work. I also tried ...
2
votes
1
answer
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How do you find the number of digits for a solution for logarithmic equation and inequalities?
In this question using the laws of logarithms to solve the equation and inequality you're not supposed to use a calculator:
You are given that $\log_{10}{4} = 0.60206$ correct to 5 decimal places and ...
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1
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Integral of Complex logarithm makes sense?
For example, I know that the principal branch of logarithm is not defined over negative real axis.
I think the integral of this logarithm along a circle doesn’t make sense.
Moreover, I know that the ...
- 455k
3
votes
3
answers
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$\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$ as a limit of a sum
Working on the same lines as
This/This and
This
I got the following expression for the Dilogarithm $\operatorname{Li}_{2} \left(\frac{1}{e^{\pi}} \right)$:
$$\operatorname{Li}_{2} \left(\frac{1}{e^{\...
- 5,337
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votes
6
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Inverse of the polylogarithm
The polylogarithm can be defined using the power series
$$
\operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}.
$$
Contiguous polylogs have the ladder operators
$$
\operatorname{Li}_{s+1}(z) ...
- 12.5k
3
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Generalizing a logarithmic inequality
Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$
$$\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{...
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Logarithms find the solution
What conditions must $a$ and $b$ satisfy for the equation to have at least one real solution? Find all the solutions of this equation:
$1+\log_b(2\log(a)-x)\log_x(b)=2\log_b(x)$
I have tried ...
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What does "any polynomial dominates any logarithm" mean here?
My textbook states that
any polynomial dominates any logarithm: $n$ dominates $(\log n)^3$. This also means, for example, that $n^2$ dominates $n\log n$
However, it wasn't clear to me what the ...
- 3,019
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Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x$
Solve the power equation $\;2\!\cdot\!3^{x^2}=6^x.$
$$2 \cdot 3^{x^{2}} = 6^x $$
$$log_3(9) \cdot log_3(3^{x^{2}}) = log_3(6^x) $$
$$2x^2 - xlog_3(6) = 0 $$
$$x(2x - log_3(6)) = 0$$
$$x = 0$$ or $$...
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Smart way to calculate floor(log(x))?
I thought of an algorithm that involves $\lfloor \log_{b} x \rfloor$ and am trying to determine its computational complexity. At first glance my algorithm looks polynomial, but I read that my ...
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Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
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