Questions related to real and complex logarithms.

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1answer
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Logarithms in Calculators?

I have no idea how to do logarithms, or even what they are, but our class recently received an extra credit problem pertaining to one. This helps me with EXACTLY what I want to do, but I have no idea ...
12
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4answers
391 views

Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$

I have no idea where to even start. WolframAlpha cant compute it either. $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$ I think it can be done with series, but I am not sure, can someone help a little so ...
0
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1answer
168 views

Stuck on derivative of logarithm of sum of exponentials

let's say that I need to calculate the following expression: $$ \frac{\partial\mathrm{log}(\mathrm{exp}(w_1 * x_1 + b_1) + \mathrm{exp}(w_2 * x_2 + b_2))}{\partial w_1} $$ How do I start? The rules ...
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2answers
50 views

stuck on logarithm of derivative of sum $\frac{\partial\mathrm{log}(a+b)}{\partial a}$

I need to evaluate an expression similar to the following: $\frac{\partial\mathrm{log}(a+b)}{\partial a}$ At this point I don't know how to proceed. $b$ is a constant so there should be some way to ...
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0answers
30 views

derivative of logarithm of function, and derivative of $x$ with respect to $log(x)$

if $y = \mathrm{log}(f(x))$ then the derivative with respect to $x$ is: $\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{f^\prime(x)}{f(x)}$ but what if I want to calculate ...
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2answers
30 views

Finding the value of an expression with logarithms

Given that $\log_{b}a=0.74$ and $\log_{b}(a-1)=0.65$ find the value of the following expression: $$\log_{b}(a^{4}-1)-2\log_{b}(a^{2}+1)+\log_{b}(a^{3}+a)-\log_{b}(a+1)$$ I tried using log laws to ...
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2answers
71 views

Explain how to get the right solution of y $dy/dx=y$

When solving the following equation to find y as a function of x: \begin{equation} dy/dx=y \end{equation} First I divide both sides by $y$ and multiply both sides by $dx$: $dy/y=dx$ Then I ...
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0answers
65 views

Evaluating a limit involving fractions and logarithms

I am trying to evaluate the following limit. Let $0 < \alpha < \infty$. Then $\begin{align*} \lim_{k \to \infty} ...
1
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1answer
48 views

Check my answer, $\lim_{n \to \infty}\frac{2^{\sqrt{\log (\log(n))}}}{(\log(\log(n)))^2\sqrt n}$

I would like someone to review my solution to this limit, the result (at least to me) is quite surprising. Assume there is a limit and it is $L$, then $$\lim_{n \to \infty}\frac{2^{\sqrt{\log ...
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3answers
117 views

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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2answers
402 views

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!
1
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3answers
87 views

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 $f(z)\equiv 0$. ...
2
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1answer
120 views

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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2answers
84 views

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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1answer
29 views

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 ...
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1answer
83 views

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$?

What is $\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz$? By letting $z = \mathrm{e}^{it}$, we get $$\int_{\lvert z\rvert=1} \mathrm{Log}(z)\,dz = \int_0^{2\pi} \mathrm{Log}(\mathrm{e}^{it}) ...
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1answer
90 views

Simplifying $2^\sqrt{\log x}$

Can this expression be simplified? $$2^\sqrt{\log x}$$ Thank you
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2answers
48 views

why is $\displaystyle \frac{\log(\sin x)}{\log(x)}$ $\quad\frac{\infty}{\infty}$ form as $x\to 0$?

In this question $\displaystyle\frac{\log(\sin x)}{\log x}$ is taken as $\displaystyle\frac{\infty}{\infty}$ indeterminate form. But $\log(0)$ is not defined so how can L'Hospital's rule can be ...
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1answer
36 views

How logarithms affect given condition

I am working with long productcs of probabilies and in order of avoinding underflow I am using the addition of (negative) logarithms. P(A) =-log(P(a1) + -log(P(a2)+.... In the end I get a positiv ...
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1answer
17 views

Logarithm Via Multiplication By Some Function

This question may be totally rubbish, I'm not sure... Basically, I'm wondering if I have some expression, say, $$f(z)A(z)$$ where $A(z)$ is some arbitraryly changing function in $z$ totally out of my ...
2
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1answer
67 views

is there any solution for $x^2 +x + 2 = e^x$ by using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
0
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1answer
83 views

Can $x^2 +x + 2 = 10^x$ be solved using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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0answers
23 views

Manipulating a logarithmic expression with ceilings

How can I simplify the first expression so that it satisfies the following inequality? $$ c(n+2) \log\lceil n/3\rceil - cn\log n \le 2\log\lceil n/3 \rceil-cn\log \left(\frac{3n}{n+2}\right)$$
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3answers
53 views

Proof of the fact that ln(a) = f '(0) for f(x) = a^x?

Looking over notes from class today and wanted to know if there is any type of proof for the fact that $\ln(a) = \lim_{h\to0}(a^h-1)/h$, which is just $f '(0)$ for any function of the form $f(x) = ...
8
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1answer
172 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
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1answer
212 views

Calculus integral evaluation using substitution

I have to find this integral: Evaluate the integral using an appropriate substitution $$\int\dfrac{8e^x+7e^{-x}}{8e^x-7e^{-x}}\mathrm dx.$$ I've tried my solution $\ln\Big[15\cdot \sinh(x) + ...
2
votes
2answers
47 views

Logarithmic function with strange bases

Given $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n$. I have rewritten $\log_{3n} 45$ as $\dfrac{\log_{4n}45}{\log_{4n}3n}$ and multiplied to get $\log_{4n} 40\sqrt{3}\cdot\log_{4n}3n = \log_{4n} ...
2
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2answers
41 views

Stuck with finding the domain of a function with a logarithm

Find the domain of the function $$g(x)=\log_3(x^2-1)$$ This is what I got so far: $$\{ x\mid x^2-1>0\} =$$ $$\{ x\mid x^2>1\} =$$ $$\{ x\mid x>\sqrt { 1 } \}= $$ I don't know where to ...
2
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2answers
135 views

Holomorphic branch of the $n^{\mathrm{th}}$ root of $f$

The problem states to prove that if $h$ is a branch of $f^{1/n}$ for integer $n > 0$ (i.e. $h(z)^n = f(z)$ for $z \in G$, $h$ continuous), then $h$ is holomorphic, where $f$ is a holomorphic ...
1
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1answer
39 views

Taking the log of both sides to determine big Theta/Omega/O

I've managed to confuse myself over this detail: Obviously: $n^2 \notin \Theta(n)$ Now if we take the $\log$ of both sides, we get: $$\log(n^2) \leq \log(cn)$$ $$2\log(n) \leq \log(c) + \log(n)$$ ...
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1answer
53 views

Principal square root of a product of complex numbers with positive real part

Given $n$ complex numbers $z_i$ with $\Re z_i>0$, why is it that $$\prod_i\sqrt{z_i}=\sqrt{\prod_i z_i}?$$Numerically, this appears to be the case, however, I don't see an easy way to prove it.
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3answers
92 views

How to solve $2\ln(x) = \sqrt{x}$ ? ln = natural logarithm

I used Microsoft Mathematics and it says $x$ is approximately $2.04\dots$ but, how do you prove it? Edit: I'm sorry if I wasn't clear enough with my question. I don't want to prove that two roots ...
0
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1answer
33 views

Differential equation $xy'+2y=0$ and the form of arbitrary constant in its general solution

If I'm solving the differential equation in the title I will get to: $$\log(y)=-2\log(x)+c$$ then I'll get $y=e^c/x^2$ eith arbitrary constant $c$. So I know I can write $y=d/x^2$ where $d$ is an ...
12
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0answers
166 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
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3answers
119 views

Solving $\log(x) = x-1$?

One can use Taylor series of the log or exp function to get the result that $x = 1$. I was wondering if there is any other simple solutions. Thanks a lot!
2
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3answers
56 views

Logarithmic Differentiation - when to use?

Sorry if this is an ignorant or uninformed question, but I would like to know when I can (or should use) logarithmic differentiation. I haven't taken calculus in a while so I'm quite rusty. So, let's ...
2
votes
1answer
71 views

Number of digits in $12^{300}$

Given: $\log_{10}2= 0.3010$ and $\log_{10}3=0.4771 $, find the numer of digits in $12^{300}$ Options: $324,323,325,\text{Other}$ Actually I tried breaking 12 into 2*2*4.. And then tried to guess ...
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1answer
40 views

Different answer when simplifying before integrating

I have been trying to get my head around this for some time now... I solve the same integral in two ways but get two different solutions. Since there can't (surely) be any sort of ambiguity when ...
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2answers
39 views

How to determine the value of a variable in a equation with powers

I'm completely rusty on this How would be the way of determing the value of x in something like this $\ 100 = \frac{50}{(1 + x)^a} + \frac{50}{(1 + x)^b} + \frac{50}{(1 + x)^c}$ a, b, c are known ...
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2answers
34 views

proof - proving a proposition involving logarithms is true or false

I'm looking at my textbook and I'm not sure how to solve this to prove whether it's true or not. (there exists x in the real)(3^x = x^2 ) Any help would be good. Thank you.
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2answers
57 views

How to find all the intersection points of the two functions $\log(x!)$ and $x$?

I am trying to find where $\log(x!)$ and $x$ intersect, and am unable to do so rigorously. I eventually have $2^x = x!$, but I am unsure how to proceed from here. Any input as to how to go about ...
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4answers
93 views

Solve $\ln(x)+\ln(x-1)=0$ for $x$

Solve the following equation for x; $$\ln(x)+\ln(x-1)=0$$ What I did is the following but I'm pretty sure its wrong.. $$\ln(x)+\ln(x-1)=0$$ $$\ln(x)=-\ln(x-1)$$ $$e^{\ln(x)}=e^{-\ln(x-1))}$$ ...
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1answer
46 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
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2answers
33 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
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2answers
94 views

How did Newton calculate 3x7 by logarithm?

This is a story about Newton I read once when I was a child. Now that book is lost and I can only tell you what I remember. When Newton was young, he had been already famous in curiosity and ...
2
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2answers
149 views

How to solve this logarithm equation

$$\log_2\left\{\log_3\left[\log_4\left(x^{3x}\right)\right]\right\} = 0$$ How would I go about solving this? I tried doing $\log_4(x^{3x}))=0$ but I don't know how to incorporate the other logs
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1answer
44 views

Calculate log of number less than raised to power

I want to calculate the value of 0.9 raised to power 17.I am using the log method. 17 * log(0.9).Am I doing this correctly?
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0answers
31 views

What does Heron's Algorithm have to do with the construction of logarithmic tables

i need a little help answering this question, what does Heron's Algorithm have to do with the construction of logarithmic tables. I know that Heron's algorithm is used for finding square roots, but ...
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1answer
41 views

Show $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$

I am trying to show that $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$ but I seem to get a conflicting result. What i did is: $n!=1*2*3*...*n \leq n*n*n*...*n=n^n$, so $\frac{n}{2} \log(n!) \leq ...
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3answers
61 views

What algorithm solves this problem? Non-linear measuring tape

A measuring tape is marked at 0, 5, 15 and 40. The distances between each mark are marked on top. At what distances should I mark 1 through 4, as well as 6-14 and 16-39? My math knowledge does not ...