# Tagged Questions

Questions related to real and complex logarithms.

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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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### Codomain of p-adic logarithm

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: $$\log(2x-1)=-x+3$$
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### Proof of a logarithm identity

I would like to know how to prove the following log identity: $x^{\frac{\log(\log(x))}{\log(x)}} = \log(x)$
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Define the following functions $\mathbb{C}\to\mathbb{C}:$ $$u(z)=\frac{\log \left(z+\frac{1}{2}\right)}{z}\quad \left[-\pi\leqslant\arg \left(z+\tfrac12\right)<\pi\right];\quad v(z)=\frac{\log z}{z}... 1answer 26 views ### During the 56th month or the 57th month? A car depreciates in value according to the model$$V=Ak^t$$where £V is the value of the car t months from when it was new. Its value when new was £12499 and 36 months later its value was £... 2answers 50 views ### Complex logarithms when computing real-valued integral My question arise when I try to calculate real-valued integral, specifically, I want to evaluate the integral $$\int_0^1 \frac{\ln \left(\frac{x^2}{2}-x+1\right)}{x} dx$$ ... 7answers 1k views ### What is “8 log 2”? [closed] When someone says "8 Log 2" what does this equate to in writing? Does it mean the following?$$ \log _{2} 8 $$And if so, what is the value of this? When I plug those numbers into this log ... 3answers 118 views ### How to solve 3(a+1)(b+1)=3^a \times 2^b? Hi I'm new to logarithms and not sure how to solve equations involving logarithms. I managed to find this equation to answer a problem solving question, however now I do not know how to solve the ... 2answers 32 views ### Graphing log with number in front of “log” When I have something like y = log_2(x) I know that I have to turn it into exponential form and get: 2^y = x. Next, I make a table for X,Y and choose about 5 values for y, typically -1, 0, 1, ... 0answers 33 views ### Order Size estimation of converging sum used for approximation of logarithm I know it can be shown that \log n=\sum_{i=1}^\infty \frac{(n-1)^i}{in^i} for \forall n\in\Re where n\ge1 For given natural m, I tried to find the order size of k = f(m,n) in order for the ... 0answers 22 views ### Proof this limit superior is finite. Let \{ w_n \} be a sequence of non-negative numbers and put M_n=\sum_{k=1}^n w_k^2 \xrightarrow{n\to\infty} \infty . Proof that$$\limsup_{n\to\infty} \dfrac{\ln \ln \sqrt{M_n \ln \ln M_n} }{\ln \...
Just a bit confused about how to evaluate the following $$\log_3 8\times \log_5 9\times \log_2 5$$ What I have done so far: I have used the change of base rule to change each log to base $3$, ...