Questions related to real and complex logarithms.
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transforming a straight band into a logarithmic spiral
I want to plot the labels and the graduations of an historical timeline onto a logarithmic spiral.
If this timeline is on the $x$-axis, $-\infty$ would project to the center of the spiral, $+\infty$ ...
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1answer
25 views
simple logarithms with exponents
Note: I am using log base 10 and I am trying to rewrite the equation using exponents instead of logs.
Here is what I have and I am wondering if I did it correctly (if not how am I suspose to solve ...
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1answer
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1answer
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Solving basic exponential equation with logs
I am having trouble with this grade 12 pre-calc question that I am sure will be elementary to most of you. I understand most of it but I do not understand one of the steps.
These are the steps in my ...
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1answer
41 views
Is this a valid way to evaluate a function based on factorials? What would be a better way?
I am working on the following factorial function:
$$f(x) = [\ln(\lfloor\frac{x}{11}\rfloor!) - \ln(\lfloor\frac{x}{12}\rfloor!) - \ln(\lfloor\frac{x}{132}\rfloor!)] + ...
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60 views
How to compute the asymptotic growth of $\binom{n}{\log n}$?
I'm interested with tight bounds for: $$f(n)={n\choose{\log{n}}}$$
It sounds like it's something simple, but I can't get a nice expression I can use.
Any ideas on how to do this?
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2answers
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Discrete Logarithm
If $p$ is a prime and $a,b$ are integers not divisible by $p$ such that $a^x \equiv b \pmod p$ with $0 ≤ x < o_p(a)$, then we define $x = L_a(b)$ and say $x$ is the discrete logarithm of $b$ ...
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1answer
38 views
Denesting Logarithmic expressions
$\log_7(\log_2(3)) + \log_7(\log_5(6)) + \log_7(\log_{11}(1/2)) = \log_7(-1) + \log_7(\log_5(3)) + \log_7(\log_{11}(6))$
This can only be simplified by using the sum to product rule and noticing that ...
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1answer
55 views
Proving that a specific gamma function is a guaranteed lower bound for a factorial function
In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that:
$$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$
I have ...
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2answers
25 views
Logarithm of a Gram matrix
Given a Gram matrix $K$, we are interested in calculating its matrix logarithm $\log(K)$, and in particular, to relate minus this logarithm to the Laplacian of a graph.
We have noticed that ...
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1answer
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Finding solutions to $\frac{2 t }{x}= B_r \log B_r + B_s + B_s x \log(B_s x)$
Given $B_r$, $B_s$, $t$ being constants, and $x$ being a variable $0\leq x\leq 1$ how can I solve this equation?
$$\frac{2 t }{x}= B_r \log B_r + B_s + B_s x \log(B_s x)$$
If i plot the two ...
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1answer
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Looking for help understanding the asymptotic expansion of the digamma function
I was recently given an example using this asymptotic expansion of the digamma function where:
$$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$
Here's the ...
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3answers
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To find the logarithm of $1728$ to the base $2 \sqrt{3}$
Find the logarithm of: $1728$ to base $2\sqrt{3}$.
Let, $\log_{2\sqrt{3}} 1728 = y$, then
$$\begin{align} (2\sqrt{3})^y &= 1728\\
2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
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1answer
33 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
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0answers
26 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
