# Questions tagged [logarithms]

Questions related to real and complex logarithms.

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### What property is being used to simplify this logarithm? $4^{log_2 n} = n^2$

I had this equation for calculating this recursion: $$T(n) = 4T(\frac{n}{2}) + n$$ I'm using a recursion tree to solve this, it's similar to this one: But anyway what matters is that from the ...
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### Applying log on a generating function

I am reading the Book SymmetricFunctions and Hall Polynomials by MacDonald and now reached the chapter "Power Sums" (Starting on Page 23). There we want to proof $[P(t)=H'(t)/H(t)]$ In the ...
1 vote
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### $\int\frac{\ln^kx\ln(1-x)}x dx$ Vs. $\int\frac{\ln x\ln^k(1-x)}x dx$

By using WolframAlpha, experimentely I observed that $$\frac1{k!}\int\frac{\ln^kx\ln(1-x)}x dx=(-1)^{k+1}\text{Li}_{k+2}(x)+\sum_{i=1}^k \frac{(-1)^{k+1-i}}{i!} \text{Li}_{k+2-i}(x)\ln^{i}x+c.$$ On ...
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### Computation of limit involving specific differences of logarithms

I wish to compute the following limit: $$\lim_{n\to\infty}\frac{1}{\ln(2)n}\left(\sum_{k=(2n/3)+1}^n \ln(k) - \sum_{k=1}^{n/3} \ln(k)\right).$$ Putting the expression into a CAS just turns out the ...
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### Iterating $\log(x\log(x\log(...)))$

For a real positive $x$, let $F(x)$ denote the sequence $$\left(\log x,\log(x\log(x)),\log(x\log(x\log(x))),\log(x\log(x\log(x\log(x)))),...\right),$$ stopping at the first nonpositive value or ...
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### Applications of the gamma function [closed]

I am a student studying logarithms and for a presentation about their uses I am demonstrating how the laws of logarithms can be used to extend the definition of x! to the reals. However, I am ...
1 vote
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### When solving Logarithmic Inequalities why are the two following methods not the same?

When solving $\log_6 \left(\frac{12x-1}{x-3}\right)<0$, I would raise both sides as exponents to a base of $6$, resulting in $\frac{12x-1}{x-3}<1$, then move the $1$ to the left side, make a ...
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1 vote
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### how to solve the following question using logarithms?

how can I use logarithms to compare a,b,c in an ascending order $$a=\frac{3^{8.7} - 3^{6.2}}{5}\\\ b=\frac{3^{11.7} - 3^{8.7}}{6} \\\ c=\frac{3^{11.7} - 3^{6.2}}{11} \\\$$ i tried to simplify a b ...
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### How much is $\log_{k}(n!)$? [closed]

How to evaluate $\log_{k}(n!)$?
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### The polynomial P(x) is equal to $(i + x)^{2024}$. Let $S$ be the sum of all the real coefficients of $P(x)$. Find $\log_{2} S$. [closed]

I think the answer to this problem is $1012$, but I'm not sure. Any ideas? Any and all help is appreciated.
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### Quick doubt on this function domain

$$f(x) = \frac{e^{1-\ln(x-x^2)}}{\ln(1 - e^{x-x^2})}$$ I was solving this, and I found the domain is the emptyset. Yet, then I checked with Mathematica and it returned me a different domain. When I ...
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### Differentiate this function: L1=ln(c11 +δλ11)+β1ln(c12 +δλ12)+Zλ11(1−n11)−αn11 ​ [closed]

Differentiate this function: L1=ln(c11 +δλ11)+β1ln(c12 +δλ12)+Zλ11(1−n11)−αn11
1 vote
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### Implication in Logarithmic Inequalities

I am trying to understand an implication I came across in a research article, and I was wondering if someone could provide more details or clarifications on the reasoning behind it. The inequality in ...
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### How to approximate the median of the numbers in the first $n$ rows of Pascal's triangle?

How can we approximate the median of the numbers in the first $n$ rows of Pascal's triangle? (The top row is the $0$th row.) Using Excel, I made a graph of the natural log of the median against $n$, ...
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### Solving $\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$ [closed]

Here's the question I came across, they're inverses in this case, but I imagine that there is a way to do that without them being inverses. $$\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$$
1 vote
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### Logarithm inequality less than 1

I'm currently working on proving the inequality $$\frac{\log(\log(m))}{\log(\log(n))} \left(1 + \frac{\log(n/m)}{\log(n) \log(\log(m))}\right) < 1$$ where $n > m$, and I suspect that using the ...
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### Calculating the Derivative of a Complex Expression [duplicate]

I am trying to find the derivative of the following expression: $y = 1 + \frac{a}{1/x - a} + \frac{b/x}{(1/x - a)(1/x - b)} + \frac{c/x^2}{(1/x - a)(1/x - b)(1/x - c)}$ I have attempted to use basic ...
1 vote
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### Solving Logarithmic Expression

In the context of the thermodynamics of mixing two separate gases at the same temperature and pressure, one has the generic equation, \begin{gather*} -\frac{\Delta S_{mix}}{nR} = X_A \ln(X_A) + X_B \...
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### Prove that $2 \int_{x}^{x+1} \log(t) dt \geq \log(x (x+1))$.

Prove that $2 \int_{x}^{x+1} \log(t) \,dt \geq \log(x (x+1))$. I want proof of this without using geometry. I need to know the techniques to solve this other than geometry. One can easily see this ...
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### Solving $\log_3x^3 - 4\log_9x - 5\log_{27}x^{1/2} = \log_94$ [closed]

Having trouble solving this simple logarithm problem: $$\log_3(x^3) - 4\log_9(x) - 5\log_{27}(x^{1/2}) = \log_9(4)$$ I’ve been stuck as I when I solve it I get an answer of $2$, when the actual ...
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### How to determine the almost accurate initial guess to calculate the $\ln(x)$ via Newton-Raphson method?

I'm developing my own Real Number class, and for that, I have to implement the Natural Logarithm function. I've used Maclaurin series of Natural log with iteration of 300 or 500. The division method ...