# Tagged Questions

Questions related to real and complex logarithms.

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### How to rigorously deduce the Laurent series of $\log\frac{z-p}{z-q}$?

Of course, the logarithm here is defined on the ring region $|z|>R\ge\max\{|p|,|q|\}$ as $$\log\frac{z-p}{z-q}=\int_{z_0}^z \left(\frac1{w-p}-\frac1{w-q}\right)\mathrm d w.$$ Here the integral is ...
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### question from logarithms [closed]

Simplify without using tables $$\frac{\log25+\log625}{\log5}$$
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### $\log(e^z - i)$ as a holomorphic function in $\mathbb{D}$

I'm learning complex analysis, specifically holomorphic functions, and need help with the following exercise: Examine if the function $\log(e^z - i)$ can be defined as a holomorphic function in ...
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### Find $\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$

I have to solve this limit: $$\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$$ I know the answer is $-\infty$. My question is, can I do this: $$\ln[\lim_{n\to\infty}n^{n}\cdot(n+1)^{-n-1}]$$ If not, how ...
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### $\log_5 {(x+1)} - \log _4 {(x-2)} = 1$

I tried to solve this equation by changing bases. $\dfrac {\log_4 (x+1)}{\log_4 5} = \log_4 (4x -8)$ $.86 \log_4 (x+1) = \log_4 (4x-8)$ Then i got stuck. Please share your idea with me.
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### Finding Limit of Nested/Continued Logarithm

For a sequence $a_n$ defined by: $$a_1 = \ln(1)$$ $$a_2 = \ln\left(\frac{1}{\ln(2)}+1\right)$$ $$\dots a_n = \ln\left(\frac{1}{\ln(\frac{1}{\ln(\dots 1/\ln(n ))}+1)}+1 \right)$$ with $n$ ...
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