Questions related to real and complex logarithms.

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0
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2answers
30 views

Rewrite a Logarithm as a product of logarithms

Can anyone help me to understand this? $$\log_2 n = \log_2e \log n$$
-2
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1answer
43 views

Difficulty finding point of inflection

The Problem is... $N(t)=\frac{200,000}{1+999e^{-0.4t}}$ Use logarithmic differentiation to find the time to the point of inflection. I know that in order to find the point of inflection I must set ...
0
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0answers
16 views

Find the complex potential given the real part $\frac{6000}{\pi}\theta$

In my 'Complex Variable Theory' notes, we have an example where we use a linear fractional mapping to solve for Laplace's equation between semi-circular plates. The Question reads: Find the ...
1
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1answer
35 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
0
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1answer
41 views

Please explain the logarithmic equation

The default equation is $(1 + x)^3=4^{-y^2} $ I solved as follows: $(-3\log_4(1+x))^{1/2}=y$ With the logarithm base equal to $4$, my idea is that $4$ is the number we have in the right part of ...
1
vote
2answers
71 views

$2\ln(-x) \neq \ln(-x^2) ? $

I know the rule $$n\ln(x) = \ln(x^n)$$ But this doesn't apply to $$2\ln(-x) = \ln(-x^2)$$ Can you see what I'm not understanding?
1
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0answers
13 views

Financial Mathematics--Finding Compounding Period given Annual and Effective Interest Rates

I'm trying to find a compounding period C when given an annual interest rate r and effective annual yield i. I'm working with the following equation: $i=(1+r/C)^C-1$ I'm having trouble re-writing ...
1
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4answers
49 views

How to prove that $(x+c)\log(\frac{c+x}{x})>c$

How to prove that $(x+c)\log(\frac{c+x}{x})>c$ for $x, c > 0$? For $\frac{c+x}{x} \ge e$ it's obvious.
1
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3answers
35 views

Limit of $f(x)=|\log x|$

My textbook solved this problem: Find $f'(1^{-})$ if $$f(x)=|\log x|$$ for the interval $x>0$ The textbook solved it by using the method described below: $$f'(1^{-})=\lim\limits_{x\to 1^{-}} \...
0
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0answers
27 views

Simplify/expand $\ln \left(\sum^n_{i=1}x_i^{\theta-1}\right)$

Can someone help simplify/expand this natural log? I want to bring the $\theta - 1$ down in front of the $\ln$ but I don't know how the rules work with the summation. $$\ln\left( \sum^n_{i=1}x_i^{\...
1
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0answers
79 views

Hard Logarithm Integral [duplicate]

I recently had this integral in my homework and don't really know how to proceed. $\int^1_0\frac{\ln (1+x^{2+ \sqrt3})}{x+1}dx$ So far I figured that $ \int^1_0\frac{\ln (1+x^{a})}{x+1}dx= \int^1_0\...
4
votes
1answer
99 views

How to compute $\int 1/x \, dx$ without knowing its anti-derivative

How do you compute $$\int\frac 1x \, dx$$ without knowing its anti-derivative to start with? Is there a way to do it by parts or substitution?
1
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0answers
41 views

Really simple question: Add $\bar4.74628$ and $ 3.42367$ .I just need to cross check answer.

Add $\bar4.74628$ and $ 3.42367$ This question is about characteristics and Mantissa.I thought my book has written the wrong answer in the example.I just wish to cross check because this seem like ...
2
votes
2answers
28 views

Natural logarithm power notation

I am trying to understand how to use Dirichlet's test for convergence and saw an example here (example 2). Show that $\displaystyle\sum_{i=1}^\infty \frac{2^{2n}n^2}{e^n\,n!}\frac{1}{\ln^2n}$ ...
0
votes
1answer
33 views

Adding logarithms with different bases

Just had an exam, this sinister question I know I did wrong lingers in my mind: Solve for $x$, $$2-\log_3(x-7) = \log_{\frac{1}{3}} (2x)$$ On phone not sure how to write the equation properly. ...
-1
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1answer
24 views

Big O notaion O(n) and logaritms [closed]

Can someone explain me the subjects Big O notation and logarithms please? I can't understand those subjects For example if I have a question like this: recall that logan is the power to which you ...
0
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2answers
54 views

Approximate $\log(1-e^x)$ where $x<0$

The title is pretty self-explanatory, I need to calculate the logit function ($x=\log(p)$): $$x-\log(1-e^x)$$ Where $x<0$, And my problem is to approximate $$\log(1-e^x)$$ I was thinking of ...
1
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1answer
73 views

Logarithm as limiting case of $n$th root

Let $f_n(x) = x^{1/n}$ where $n \in \mathbb N$, and let $g(x) = \log(x)$. We can compute $f_n'(x) = \frac{1}{n}x^{-1 + \frac{1}{n}}$ and $g'(x) = x^{-1}$. Let's define $f_\infty(x) = \lim_{n \...
1
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2answers
45 views

Proof of log 2 base 10 value

Is there a way to prove log 2 base 10 <= 0.301 other than verifying the value using a calculator? Please give a detailed explanation, if proof is possible.
0
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1answer
36 views

How to solve the logarithmic equation $\ln(x + 4) = 6$?

I would like to learn the steps for solving this math problem. One of my classmates gave me this problem, and I need help solving it. $\ln(x+4)=6$
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1answer
59 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
-1
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2answers
45 views
0
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2answers
33 views

How to simplify the expression $(\log_9 2 + \log_9 4)\log_2 (3)$

Our test asked to simplify $(\log_9 2 + \log_9 4)\log_2 (3)$. I simplified the first parenthesis to be $\log_9 (8)$. So, now I have $\log_9 (8) \cdot \log_2 (3)$ and I can change to base $10$ and ...
0
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1answer
26 views

basic math problem relating to log thought my answer is not matching with the alternative

$1/2 \log c = 0.915$. Calculate $c$. It is a basic math problem but my answers are not matching with the alternatives. $1/2 \log c= c^{1/2}$ $ c^{1/2}= 0.915$ $c = 0.915 \times 0.915=0.83448$ ...
1
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1answer
14 views

Verify whether or not expression is true or not for z>0

I need to verify whether or not the below expression is true or not for $z>0$. I'm trying to understand the rules of logarithms but I can't figure out how to apply it myself or where to even begin. ...
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2answers
60 views

Solution of $5^{\log x}+5x^{\log 5}=3$

Solve for $x$ $$5^{\log x}+5x^{\log 5}=3$$ where base of log is $a$, $a>0$ and $a \neq1$ Could someone hint as how to initiate this question? I am not having any idea as how to proceed.
2
votes
1answer
71 views

How to prove $({\log_2 x})^{n+1} \le x^n$

I want to show that $({\log_2 x})^{n+1} \le x^n$ when $n \ge 1$ and $x \ge 1$. I know that ${\log_2 x}$ can be shown to be $\lt x$ with: $x \lt 2^x$ $\log_2 x \lt x$ and obviously adding the same ...
1
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1answer
32 views

How are floating-point numbers logarithmically distributed?

From what I remember from a lecture I had of a course I'm attending called "introduction to computational science", floating-point numbers are distributed logarithmically. What does it mean? And how ...
1
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2answers
89 views

How to solve the equation $x \log \log x = n$

I would like to solve the equation $x \log\log x = n$. I've seen a lot of post about the equation $x \log x$ but here I have a composition of $\log$. How can I solve it ? Thank you very much.
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0answers
30 views

Nested logarithms to represent real intervals.

Out of curiosity from this question regarding writing real numbers as an iterated sum/difference of square roots I started experimenting with another family of functions: $$\log[ \exp[1] \pm (\exp[1/...
12
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2answers
208 views

What is $\int_0^1 \ln (1-x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx$?

There are well-known closed-form evaluations for integrals of the form $\int_0^1 a(x) \ln \left(\ln \left(\frac{1}{x}\right)\right) \, dx $ for certain algebraic functions $a(x)$. For example, an ...
0
votes
1answer
33 views

When will the population of a sample double (using dif-eq)?

I have the initial equation $$\frac{dP}{dt}=kp$$ where P is the population, t is time, and k is some positive constant. The rest of what I'm given is that P(0) = A, what is the time for the population ...
0
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2answers
15 views

Solution to initial condition problem

$y=-ln(1-e^{(t+c)})$ I'm trying to find the solution to the initial condition $y(0)=-ln2$ Isolate c $0=ln(2)-ln(1-e^c)$ $0=ln({2\over1-e^c})$ $-e^c=2-1$ $e^c=-1$ $c=0$ I can't figure out ...
0
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0answers
20 views

Weighted logarithmic ranking

I want to have a ranking of players by percentage of shots made, weighted by the total number of shots attempted. The weighting should follow a log scale, so for example Player A has 100% accuracy, ...
0
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1answer
28 views

What is the argument of the logarithm operator called?

In the expression $ln(y)$, what is '$y$' called. I'm asking for a noun analogous to exponent in $x^n$, where '$n$' is called the exponent. If I'm not mistaken, '$x$' in this case is the radix, or ...
1
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1answer
16 views

Is it possible to clear the x using the Lambert function?

$ y = \frac{x^2}{4} - \frac{ln(x)}{2} $ Solving, I get to: $ e^{4y} = \frac{e^{x^2}}{x^2} $ But I don't know how to continue.
0
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1answer
46 views

Stuck solving $\ln(e^y-1)-y=t+c$ for $y$

I'm trying to solve for $y$ $\ln(e^y-1)-y=t+c$ $e^y-1=e^{(t+c+y)}$ $e^y=e^{(t+c+y)}+1$ $y=t+c+y+1$ Where am I going wrong?
0
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2answers
54 views

Tricky Logarithmic inequality

I have tried proving this logarithmic inequality but I did not succeed. I tried to put every term on one side, I expanded and tried to use one of the properties of logarithms but the proof does not ...
1
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0answers
45 views

Question about the connection between exponential and logarithmic functions

Does this make sense to anyone? What advice would you give me to clarify my reasoning and explanation? One of the really "neat" features of the exponential function: $$f(x)=e^x$$ is the fact that ...
0
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2answers
50 views

Lambert W function with natural log

I need to solve the next equation x: $d-x+yln[\frac{d}{x}]=b$ I inserted this into Wolfram Alpha and it returned: $x = y \Bbb{W}[\frac{e^\frac{d-b}{y}d}{y})]$ y, d, b, and x are all real, ...
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3answers
44 views

Sieve of Eratosthenes Time Complexity Clarification

I've found plenty of sources claiming that the time complexity of the prime sieving algorithm Sieve of Eratosthenes is $O(n\log(\log n))$ where $n$ is the input. However, is this $\log_{10}$ or $\ln$? ...
0
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1answer
24 views

Is it possible to simplify $y=100x\cdot\log_{x+1} 2$ (Solved)

Is there any way to simplify the following equation, or any way to reconfigure it in a way that is possible to graph? $$y=100x\cdot \log_{x+1} 2$$
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0answers
13 views

Calculating Split Info with given equation not matching solution

I'm given the formula to calculate the Split info but cannot seem to calculate the correct answer of 0.926 that the example shows. Split Info $= -\Sigma \frac{\mid D_j\mid}{\mid D\mid} * log_2 (\frac{...
1
vote
1answer
56 views

How to solve $y= x \cdot 2^x$ for x

Can anyone help me solve this for $x$: $y= x \cdot 2^x$ I know for $y= 2^x$, that $\log_2(y) = x$ And I can get $\displaystyle \frac yx = 2^x \implies \log_2 \frac yx = x $ But I can't ...
0
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0answers
26 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
0
votes
1answer
22 views

Decomposition of the entropy

So, I'm reading about this property in the MacKay book. But I don't fully get it. Can someone explain it to me? There's this example: A source produces a character $x$ from the alphabet $A = \{0, ...
0
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2answers
41 views

Computing first k digits and last k digits of a large number using logarithm

How do we compute the first $k$ digits and last $k$ digits of a large number say $2^{N-1}$ for bigger values of $N$ using logarithms? An example for the algorithm will be greatly appreciated. I got ...
1
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0answers
13 views

Deformations in Variational Bayesian method

I'm studying Topic Model, but I can't understand the following transformations. $F$ is variational lower bound. $$\begin{eqnarray} F[q(z_{1:n}, \phi, \pi)] &=& \int \sum_{z_{1:n}} q(z_{1:n}) q(...
3
votes
1answer
82 views

$x+\ln(x)=0$, what is $x$?

My friend came across this strange equation and I cant find mathematical way to find $x$ without drawing $x$ and $-\ln(x)$ and see that they come across at almost $x=0.5$. Can any one help?
2
votes
1answer
28 views

Solving Weird Logarithms without a Calculator

Given "$x = \log 8$", it is very easy to rewrite the expression as "$10^x = 8$", which cannot easily be solved for by hand. However, if I plug "$x = \log 8$" into my calculator, I get "$x = 0....