Questions related to real and complex logarithms.

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0answers
35 views

How do I show that the integral $\int_0^\infty x^{-a} |\log x|^b dx$ only converges when $a = 1$ and $-2 < b < -1$?

This came up in a previous question, but was closed because the question wasn't terribly clear. I don't want to edit the other question substantially because it's not mine so I'm asking a new one and ...
4
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5answers
54 views

Solving logarithmic equations including x

Let $$\log_3(x-2) = 6 - x$$ It's obvious drawing the graphs of the two functions that the only solution is $x=5$. But this is not really a proof, rather than observation. How do you prove it ...
3
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2answers
57 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
2
votes
3answers
42 views

Logs rules and Solving

I've got the equation : $$-1=\frac{-8e^{-t} + 3e^t}{2e^t}$$ I've moved some stuff around to get : $5e^t = 8e^{-t} $ But not sure where to go from here. Thanks for any help
2
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3answers
42 views

Set of real $a$ so that the inequality is defined but isn't true for a real $x$

$$x(x-\sqrt {4+\log_a7})\lt \log_7 \frac a{49}$$ I reach the interval $(0,1)$ after looking for the discriminant of the quadratic to be less than zero. However, the solution in the book is an ...
2
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1answer
44 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
1
vote
1answer
18 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
1
vote
1answer
58 views

Derivative of $(\ln x)^e$ [duplicate]

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ...
2
votes
1answer
31 views

Confused with integral and natural logarithm

When reading about ideal gas and adiabatic expansion, I got stuck with the following: $$W_{ab}=\int_{{\it V_a}}^{{\it V_b}}\!\,{\rm ...
1
vote
2answers
48 views

How to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$

I want to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ if $p > 1$ and $r \in \mathbb{N}$. To show this, I wanted to use that $\lim_{x \to 0} x \ln x = 0$, and in fact if $p \geq r$ we can write ...
0
votes
1answer
15 views

Integer solutions to the inequality $\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$

$$\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$$ If $x$ is of the interval $[-8,10]$ Now I solved this, tried to limit $x$ as much as I could but I consistently get that there should be $10$ values of $x$ ...
2
votes
1answer
73 views

Solve for $x$: $x =\ln(x)^4$

I plotted the functions on both sides and it shows the equations has at least three solutions. Is there some non-interative (not sure if i used this term correctly - i mean the way you would solve, ...
0
votes
1answer
16 views

Question about a simple rule of the complex logarithm

According to the Wikipedia page on complex logarithms: Also, the identity $\log(xy) = \log x + \log y$ can fail: the two sides can differ by an integer multiple of $2\pi i$. Does the same hold ...
4
votes
1answer
47 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
0
votes
2answers
39 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
1
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0answers
31 views

What properties does $a(n)$ have to fullfill to get $\log(2^{cn}+a(n))\sim\log(2^{cn})$?

Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...). Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, ...
0
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1answer
35 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
-1
votes
2answers
26 views

Number of digits

I've been trying to find a solution to this problem for a while but I just can't seem to find the connection between the numbers and I really need help. I apologize if a problem like this one has ...
1
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1answer
13 views

To find, wether '1' lies in the range of f, where $f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$?

$f(x)=[ln(\frac{7x-x^2}{12})]^\frac{3}{2}$, For the given function, the question is whether, f(x) can equal 1 for some real value of x?
0
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0answers
23 views

Multiple polylogarithms

$1)$ If $$I(a_o;a_1 \dots a_n ; a_{n+1}) = \int_{a_o}^{a_{n+1}} \frac{dt}{t-a_n}I(a_o;a_1\dots a_{n-1};t)$$ and $G(a_1 \dots a_n;z) = I(0 ; a_n \dots a_1;z)$, where $G$ is a multiple polylogarithm, ...
1
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3answers
200 views

How to find the inverse of $n\log n$?

So I'm on chapter $1$ of introduction to algorithms & at the end the book proposes a problem: here The answers are there & I was able to work through most of them myself despite my lack of ...
0
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1answer
16 views

The domain of logarithmic functions with sinx and cosx

I need to solve this equation: $\log_{\cos x} \sin x + \log_{\sin x} \cos x=2$ But in order to solve it, I first need to find the domain. What I did was this: $\cos x\neq1 \wedge \sin x\neq1 $ ...
0
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1answer
33 views

Polylogarithms and the shuffle algebra

$1)$ Write $\text{Li}_2(1-\frac{1}{x})$ in terms of $\text{Li}_2(x)$ and logarithms by considering its integral representation and suitable changes of variables. Attempt: The di-log is defined as ...
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votes
2answers
41 views

Logarithm with nth root [closed]

I made it but the result is very strange. I want every step to the result $$ \large 6\log_{10}\frac{\sqrt2}{\sqrt[3]{3+\sqrt5}} $$
7
votes
4answers
489 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$ ...
1
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1answer
25 views

tight estimate for a log-linear inequality

Given $q>0$ and $p$, how do we get a tight estimate for the smallest $x$ such that $x\log(x)+px \geq q$? (such an $x$ always exists).
2
votes
1answer
96 views

Are logarithms radicals? [closed]

Does the set of all logarithms with a radical base and argument belong to the set of all radicals? A simple yes, no answer will suffice, an explanation would be wonderful. EDIT 1 Can a logarithm with ...
1
vote
2answers
80 views

derivative of $\ln(4)$

what is the derivative of $\ln(4)$? I am trying to find the derivative of this equation: $h(x)=\ln(\frac{x^3\cdot e^x}{4})$ by rules of logs I simplified the $h(x)$ to the following: ...
2
votes
3answers
53 views

Values of a for which equation $\log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert$ has a unique solution

\begin{equation*} \log_ax = \lvert x+1 \rvert + \lvert x-5 \rvert. \end{equation*} I don't even know how to approach this one, any hints would be amazing. I tried separating into two cases, where ...
1
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1answer
24 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
0
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2answers
60 views

Solutions of $2^x 7^{1/x}\le 14$

The solution is supposed to be $(-\infty,0)$ and $[1,\log_2 7]$. What I get when solving the problem is $(-\infty, \log_2 7]$. Where did I get it wrong? I start by dividing both sides by 14, then ...
0
votes
1answer
50 views

logarithmic Series

I'm aware that by properties of logarithm $$\sum_{k=1}^n \ln (k) = \ln (n!)$$ My question is if $$\sum_{k=1}^n \ln^2 (k) = \ln^2 (n!)?$$ Because when I am verifying the value where $n = 5$, I get ...
0
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0answers
26 views

Discrepancy on the standard deviation of logarithmic function

Good day, Sir/Madame! I'm currently working on the standard deviation of a particular function $\frac{2}{\pi} \ln n$, where n is the degree of certain random polynomial. By the use of computer ...
2
votes
1answer
30 views

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?

Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$? By the way, I do not know how the adverb "explicitly" is strictly ...
0
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2answers
25 views

SUmmation of natural logarithm [duplicate]

Good day! Is there a formula that approximate the summation of natural logarithm of N as N runs from 1 to infinity?
0
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0answers
17 views

Estimation for a logarithmic function in $(0,\,1)$. A series should be used?

Let $f(t)\geq C_1t^{-\alpha}$ for all $t\in(0,\,\infty)$ and for some $C_1>0,\,\alpha>0$. and let $g(t)\geq C_2\left(\ln(t^{-1})\right)^\beta$ for all $t\in(0,\,1)$ and for some ...
1
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0answers
41 views

Solve $x=C \log(C \log(x+A)+B)$

Is it possible to resolve an equation of the type $$x=C\log{(C\log{(x+A)}+B)}$$ (where $A$, $B$, and $C$ are real-valued parameters) for $x$? As far as I can see, the function on the right hand ...
0
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1answer
49 views

Logarithm problem

If $a^x=b^y$, then how come $x\log a=y\log b$ holds? Can anyone show me how this is with all steps and necessary logarithm formula?
2
votes
2answers
63 views

what will be the value of this integral

$$ \large{ \int^{\Large{\frac{\pi}{2}}}_{0} \left[ e^{\ln\left(\cos x \cdot \frac{d(\cos x)}{dx}\right)} \right]dx}$$ We know that $\large{a^{log_a(c)} = c}$. But in this question, the expression in ...
2
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1answer
48 views

Is this manipulation with logs allowed?

$$\left( \frac{6}{7} \right) ^n < \frac{1}{65}$$ The answer is, by looking at which way the sign should be round: $$n > \log_\frac{6}{7}{\left(\frac{1}{65}\right)} \implies n>\frac ...
2
votes
3answers
80 views

L'Hôpital's rule exercise with natural log function

I'm looking for some advice on the following exercise: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x$$ This is my work so far: $$\lim_{x \to 0^+}{\ln{(\frac{1}{x}})}^x = \lim_{x \to ...
1
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1answer
16 views

Compound Interest Calculation

In __________ years a sum will double at $5\%$ per annum compound interest. Options given are: a. 15 years 3 months b. 14 years 2 months c. 14 years 3 months d. 15 years 2 months The way to ...
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0answers
17 views

Troubles understanding task for complex logarithm.

I have troubles understanding this question and what to do, the goal is to show that there is no complex determination of the logarithm and square root and those two are just some parts of the whole ...
6
votes
3answers
90 views

Solve $6^{x+8} = 4^{x-1}$

I tried doing $log_6\left(6^{x+8}\right) = log_6{4^{x-1}}$ I got stuck, and I don't think that was the right route.
1
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1answer
30 views

Upperbound a logarithmic expression that has a covariance matrix

Let $\Sigma$ be a $2\times 2$ covariance matrix and ${\bf h}$ a vector of complex values entries. $$A= \log(1+ {\bf h}^* \Sigma {\bf h} )$$ $$\Sigma = \begin{bmatrix} 1-|\rho_1|^2 & \rho_3 - ...
3
votes
2answers
145 views

Basic Logarithm question - I can't get both answers from quadratic

Here's the Question : If $xy$ = $64$ and $\log_x y + \log_y x = \frac{5}{2}$, find $x$ and $y$ I can get this to $$log_x y + \frac{1}{\log_x y} \frac{5}{2}$$ let $\log_x y = N$ $$N + ...
0
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1answer
65 views

$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$ [duplicate]

$2^{x^{\cos(x)}}\sqrt{\cos(x)}$ can you rearrange mathematically to ${\cos(x)}\sqrt2^{x^{\cos(x)}}$ if $x > 0$ and $\cos(x) > 0$
2
votes
1answer
86 views

Integral with Logarithms

$$\displaystyle \int _{ 0 }^{ \pi /2 }{ \log(\cos(x))\log(\sin(x)) \ dx } = \dfrac { \pi { \ln}^{ A }(B) }{ C } -\dfrac { { \pi }^{ D } }{ E } $$ $$$$ This was one solution, but it went completely ...
0
votes
1answer
46 views

Proof of $\log^x{x} > x^{\sqrt{x}}$ for big $n$

How can I prove, that $$\log^x{x} > x^{\sqrt{x}}$$ for big $n$ ? I tried to logarithm those expressions, deduct them, somehow estimate the values but no luck. After few tries, I ended up with ...
2
votes
4answers
44 views

Gradient of a curve $y=\ln \sqrt{x+y}$

Find the gradient of the curve $y=\ln \sqrt{x+y}$ at the point when its y-coordinate is 1. My attempt, I differentiated and I got $\frac{dy}{dx}=\frac{1}{2x+2y-1}$. But I've problem in finding the ...