Questions related to real and complex logarithms.

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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
32 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
votes
1answer
22 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
votes
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
vote
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
vote
2answers
36 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
votes
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. $\ln(x+4)=6$
-5
votes
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
votes
2answers
45 views
0
votes
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
votes
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
vote
1answer
13 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. ...
1
vote
2answers
53 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
vote
1answer
28 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
vote
2answers
86 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.
1
vote
0answers
28 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 ...
11
votes
2answers
183 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
votes
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
votes
0answers
18 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
votes
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
vote
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
votes
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
votes
2answers
53 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
vote
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
votes
2answers
48 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, ...
0
votes
3answers
43 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
votes
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? $$y=100x\cdot \log_{x+1} 2$$
0
votes
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 ...
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
votes
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
votes
2answers
24 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
vote
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}) ...
3
votes
1answer
78 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
votes
2answers
15 views

find $x$ from logarithm expression for $\log_{10}$ fraction

$$\log_{10} x = 0.5$$ I know if $\log_{10} x = 2$ then $x$ is $100$ but I don't know how to work out for a non obvious answer.
0
votes
1answer
50 views

Domain of $\ln\left(\frac{6}{6+x-x^2}-1\right)+\arcsin\left(\frac{x+1}{3}\right)$

blob:https%3A//mail.google.com/ea67134d-45a0-4cc0-9ec7-abf6d5a50852 I believe that my first condition is wrong but I don't understand why. Can somebody please help?
0
votes
2answers
39 views

Can $\ln|\cos x|$ be written as $-\ln|\sec x|$? absolute function

$\ln|\cos x| = \ln|1/\sec x| = \ln|(\sec x)^{-1}|=-\ln|\sec x|$ Is what I am doing valid? Or is it not correct because of the absolute function?
0
votes
2answers
71 views

$\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$; What are the differences? [closed]

What is the difference between the following: $\log^2n$, $\log n^2$, $\log \log n$, $(\log n)^2$
3
votes
6answers
120 views

Why isn't $-2$ solution for $x$?

I came across an logarithm problem recently. I don't know why solution to this problem cannot be $-2$. Now, don't downvote now because you don't know why I'm asking this. I know that logarithms' ...
0
votes
1answer
27 views

Argument principle and the principle branch of the complex logarithm

I've just been reading about complex analysis and came across the Cauchy argument principle. In my understanding you are taking the contour integral of $\frac{f'(z)}{f(z)}$ around a designated path. ...
2
votes
2answers
69 views

What is $\log(0/x)$?

$\log(a/b) = \log(a) - \log(b)$; Is $\log(0/x) = -\log(x)$? I watched a video claiming $\log(1/x) = -\log(x)$, which I get because $1/x = x^{-1}$ and $\log(x^y) = y(\log(x))$ but $\log(1)$... I ...
1
vote
4answers
70 views

relation betwn ln and e

If $f(x) = ln(x)$ and $f^-1(x) = e^x$ then is $e^x = 1/ln(x)$??? because I see $e^9 = 8103$ but $1/ln9 = .455$ How are they reverse? I don't understand!
0
votes
2answers
28 views

Log value to absolute

I am confused how to convert from log value to absolute value from the graph. Below is an example: In the graph, it shows the correlation of age of week and weight of placenta (in log). I can get ...
13
votes
3answers
2k views

What is the difference between the three types of logarithms? [closed]

In complex analysis I came across three types of logarithms namely $\ln$, $\log$ and $\text{Log}$. What is the difference between the three?
2
votes
1answer
68 views

How can you solve for s in this very complex problem?

I recently stumped across a problem, which I need to solve. Of course, I used an calculator and I got $s=3$, but I want to know how to do it step by step. The problem is kind of complex: ...
3
votes
2answers
98 views

Solve $\sqrt x = 1 + \ln(3 + x)$ algebraically

I am having trouble with this homework problem. I am able to graph and find the solution, but I am curious as to how one would do this algebraically. The way I began, was subtracting $1$ on both ...
0
votes
0answers
24 views

Evaluating right hand limit for a function

I want to prove the following right-hand limit (one sided limit) using $\epsilon-\delta$ definition; $\lim_{u\to 0^+} {u^{s_0} f(-ln u)} = 0$ where $f$ is a function from $R \to R$ and $s_0$ is a ...