# Tagged Questions

Questions related to real and complex logarithms.

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### rules for evaluating powers of logarithms

What rules are we using to show that $3^{-s}=\frac{1}{2}$ if $s=\frac{\log 2}{\log 3}$ I cannot understand how you can raise a number to a logarithm divided by a logarithm
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### What is the value of x? Related to Indices.

Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but ...
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### how to solve $\log{x}=cx^4$ for $x$

I was wondering if there is a general solution for this form of equations: $$\log{x}=cx^4$$ Tried: $$x = e^{cx^4}\\ xe^{-cx^4}=1$$
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In this example the value of the integral \begin{align} I_{3} = \int_{0}^{1} \frac{\ln^{3}(1+x)}{x} \, dx \end{align} was derived. The purpose of this question is to determine the value of the more ...
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### What is the meaning of $\log^2n$ and how should it be read in word form?

$\log^2n$ is what I need assistance with. How is this read in word form? What exactly does this mean? No matter how much I read about logarithms, they still seem new to me.
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### Prove that a logarithm is irrational [duplicate]

I’m stuck with the following problem: Prove that $\log_{2} 3 \in \mathbb{R} - \mathbb{Q}$ . Thanks in advance!
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### Show that a certain entire function is identically zero

Let $f(z)$ be an entire function satisfying $$\left|\,f\left(\frac{1}{\log(n+2)}\right)\right|<\frac{1}{n}, \quad\text{for all n\in \mathbb N}$$ for $n\in \mathbb N$. Show that $f(z)\equiv 0$. ...
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### A closed form for $\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1)$

This is another 'arctanlog' series: $$S=\sum_{n=1}^{\infty}(-1)^{n-1}\arctan\left(\frac{1}{n}\right)\ln(n^2+1)$$ Maybe differentiating with respect to some parameter could be of interest. What ...
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### Any power of logarithm is $O(N)$

This is more of a computer science question but it uses calculus and proof techniques so I think it might be more appropriate here. Basically, how do I prove that, for any constant $K \geq 1$, ...
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### Cosine of the Natural Logarithm - Series Expansion

I am interested in a computable series expansion of the following equation: $f(n) = \cos(\log(n))$ Specifically, I am interested in real values of $n$ where $n>1$. From basic series definitions ...