Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,424 questions
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Complex logarithm base 1
Is a logarithm with base 1 defined in the field of complex numbers? I have not found any information about this. In real numbers, this is uncertain because $ \ln(1) = 0 $ and
$ \log_a(b)= \frac {\ln(...
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Question about $\lim_{n\to\infty} \ln(n) =\infty$ [closed]
So I know that $\lim_{n\to\infty} \ln(n)=\infty$; I've seen some proof online using the mean value theorem. But is it not easier to assume that it converges, so that $\lim_{n\to\infty} \ln(n)=k$ where ...
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Prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$ where $n$ is a natural number.
How can I prove that $\sum_{k=1}^{n} \left\lfloor \log_{2}\!\left(\frac{2n}{2k-1}\right) \right\rfloor = n$, where $n$ is a natural number?
I discovered this identity while trying to prove Prove using ...
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Most general Logarithm of a matrix
Consider any matrix $A \in \text{GL}_d(\mathbb{C})$, i.e, a square invertible matrix. We define a logarithm of $A$ as any matrix $X$ such that $$e^X = A.$$
Our objective is to find of possible ...
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Confused about the definition of branch of complex logarithm
There is something about the branch of complex logarithms that I do not understand correctly... Any clarification would be appreciated!
Let $f(z) = z^2$ and let $\Omega = \mathbb{C} \setminus \mathbb{...
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Expected value of the logarithm of a sum of exponentially-growing values.
I have a problem related to the idle game Clicker Heroes, in finding the expected $\log_{10}$ value of the rewards gained from defeating bosses throughout an ascension. For boss number $k$, the reward ...
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What happens with a log function when its base is equal to another function?
I have a problem stemming from looking into ways to approximate the Lambert W function on a graphing program like Desmos. In my process of graphing functions, I came across a question I never had ...
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Why is the natural log added to this differentiable function?
I am refreshing my calculus knowledge using. The workbook direction for the problem is:
Perform the following derivative, where:
$$\cot(4 \theta^2 - 1) > 0.$$
The author then presents the below ...
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How to reformulate the log function as its equivalent conic programming?
I am trying to solve an optimization problem that contains power functions. I reformulated the problem via a logarithmic function, and it works well. The terms of the problem involved are similar to $\...
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Find $ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$ [closed]
I see the proccedure of:
How I find the limit of $\frac{2^n}{e^{p(n)l}}$
I didn’t understand how it applies in my case:
$$ \lim_{x\to\infty} \frac{3^{x}}{e^{x}}=+\infty$$
$$ \lim_{x\to\infty} \frac{\...
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2
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Exponential and polynomial problem
I cannot find a closed form solution for $x$ in $\dfrac{x^2 e^x}{e^x - 1} = k$ where $\{x,k \} \in \mathbb{R}^+$. I thought there might be a PolyLog solution, but apparently there isn't.
Is there ...
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Limit of $\sum_{k=1}^{n-1} \frac{1}{2^n}\binom{n}{k}\log_2\!\left(\frac{n}{k}\right)$
I am trying to prove the subsequent statement, but I did not make any progress as of yet.
$$\lim_{n\:\!\to\:\!\infty} \,\sum_{k=1}^{n-1} \frac{1}{2^n}\binom{n}{k}\log_2\!\left(\frac{n}{k}\right) = 1$$
...
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Determine the smallest possible value of the natural number $ a_1$
Determine the smallest possible value of the natural number $ a_1$, knowing that there exist natural numbers
$ a_1 \geq a_2 \geq \ldots \geq a_{100} \geq 2 $ with the property that
$$
\left\{ \sum_{k=...
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Is $\log_0(0)$ undefined, indeterminate, or both? [duplicate]
I understand that $\log_1(1)$ is considered an indeterminate form, but the expression $\log_0(0)$ seems even more subtle. Algebraically, it is undefined because a logarithm cannot have a base of zero, ...
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interpolation type inequalities for log or certain convex functions
We have the following inequality for the logarithm: given any $0<a<1$, there exists a constant $C_a$ such that $$\log (1+x) \leq C_a \frac{x}{(1+x)^a} $$ holds for all $x>0$. In other words, ...
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