# Tagged Questions

Questions related to real and complex logarithms.

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### Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
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### $\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
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### Interpreting $\log_2\left(\frac{1}{0}\right)$ and $\log_2\left(\frac{0}{0}\right)$

I'm hoping someone can confirm what I've done is correct. I am working with biological datasets ... about 100 RNA-Seq datasets and I'm trying to analyze the relative up or down regulation of genes. ...
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### Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
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### What are some methods to show $\log$ is not a rational function?

It is easy to show $\log$ isn't a polynomial (no continuous extension to $\mathbb{R}$). More challenging is showing it isn't rational. Suppose it were a rational function. Then write, the fraction ...
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### How do I simplify this Log with a Fraction in it?

So I have: $$\log_2(5x) + \log_2 3 + \frac{\log_2 10}{2}$$ I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first ...
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### Find how many years must elapse before the proportions of red kangaroos and grey kangaroos are reversed, assuming the same rates continue to apply.

I have this question (sorry I'm not able to embed it): Q.7. There are approximately ten times as many red kangaroos as grey kangaroos in a certain area. If the population of grey kangaroos ...
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We have the natural map $$\log: \mathbb{C}^\times \to \mathbb{R}$$ $$z \to \log |z|$$ Is there a p-adic analogue of this? By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, ...
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### It's correct to say that $\log _{ 1 }{ 1 } =1\quad$ & $0$? [duplicate]

$\log _{ 1 }{ 1 } =1\quad$ v $0$? Because $1^1 = 1$ and $1^0 = 1$?
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### Solving for x with logarithms

I've been asked to solve for $x\,$ in $5^x + 4·5^{x+1} = 63$ The answer is $x = \frac{\log3}{\log5}$ I cannot do this without a calculator. Is there a particular method I should be using ...
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### How to prove $f(x) = \ln x$ continuous by proving first that $f(x)$ continuous at $1$, and then by using $\ln (xy) = \ln(x) + \ln(y)$. [duplicate]

I have a question concerning the proof of the continuity of $f(x) = \ln x$. I read in a comment by Pedro Tamaroff to ncmathsadist's answer to this question that this can be proved in two steps: ...
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### How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...
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### Summation Closed form for floor$\left(\log_n\right)$

The closed sum for the floors of logs of consecutive integers is: $$\sum_{i=0}^n \lfloor \log_2i\rfloor = n\lfloor \log_2n\rfloor-2^{\lfloor \log_2n\rfloor+1}+\lfloor \log_2n\rfloor+2$$ This formula ...
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### Principal argument summation

Let $\text{Arg}$ be a an principal argument in $(-\pi, \pi]$. I know that, for all $z_1,z_2\in\mathbb{C}\setminus \{0\}$, the expression $\text{Arg}(z_1z_2)= \text{Arg} z_1 + \text{Arg} z_2$ doesn't ...
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### Log equation $\log(2x-1) = -x+3$ with two non log values [closed]

What is the correct approach to solving a log equation with more than one non log value? Please demonstrate using the following equation: $$\log(2x-1)=-x+3$$