Tagged Questions

Questions related to real and complex logarithms.

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0
votes
1answer
14 views

How to deal with such inequalities?

I have that $$ Y \geq n e^{- 1- t \log t + o(1)}$$ and $$Y \leq n e^{\log n +t - t \log t}.$$ Now I would like to find values $t_0(n)$ and $t_1(n)$ such that $$Y \rightarrow 0 \text{ for all } t ...
0
votes
1answer
31 views

Efficient ways to evaluate an integral with a logarithm

Is the approximation in terms of series for the logarithm $$\log(z)= \sum_{n=0}^{\infty}\frac{2}{2n+1}\Bigl(\frac{z-1}{z+1}\Bigr)^{2n+1} $$ a good approximation if I replace this series inside the ...
0
votes
0answers
22 views

Proving an identity from a dilogarithm function. [duplicate]

If $\def\Li{\operatorname{Li}}\Li'_{2}(z) = - \displaystyle\frac{\ln(1-z)}{z}$, how does one get the identity, $$ \Li_{2}\left(- \frac{1}{z}\right) + \Li_{2}(-z) + \displaystyle\frac{1}{2}(\ln(z))^2 ...
-2
votes
0answers
44 views

Why is the natural logarithm 'natural'? [duplicate]

Simple question: Why is it that the natural logarithm is called 'natural'?
6
votes
2answers
28 views

Deriving the analytical properties of the logarithm from an algebraic definition.

Definition: The base $a$ logarithm ($a\in]0,1[\cup]1,+\infty[$) is the continuous function defined by: $\log_a(xy)=\log_a(x)+\log_a(y)~~\forall x,y>0$ and $\log_a(a)=1$ If I used this definition ...
6
votes
5answers
506 views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
1
vote
1answer
58 views

Show that $\log \log z$ is analytic

Show that $Log( Log z$) is analytic in the domain consisting of the $z$ plane with a branch cut along the line $y = 0, x ≤ 1$. As of now im not too sure on how to solve this problem, so i was ...
0
votes
1answer
18 views

How do I solve $0 = x\times114 - x\times\log_3(x) - 20.28\times y$ in matlab for different values of $y$?

I have $y = 10^3, 10^6, 10^9, 10^{12}, 10^{15}, ...$ and above mentioned equation. How do I solve (i.e. getting values of x for different y) and plot this equation in MATLAB ?
2
votes
3answers
77 views

Solving $\ln(x) = e^{-x}$

I'm trying to solve $\ln(x) = e^{-x}$ but I can't really get how to do it :( (Removing a statement that was incorrect, as explained by the comments below) Additionally, while I started to solve it I ...
0
votes
0answers
11 views

maximization of a function with random variable

I would like to know whether this is true in general, and if not when this can be. I am not sure and so I am mostly asking for confirmation. So, is the following correct ? $$\log [\max_{x} ...
1
vote
2answers
35 views

Difference between `log n` and `log^2 n`

I'm researching the different execution time of various sorting algorithms and I've come across two with similar times, but I'm not sure if they are the same. Is there a difference between ...
2
votes
4answers
37 views

Solving inequality with logarithms.

I was playing around and found this $$x\log a\le a-1$$ Solve for $x$ in the above equation, where $a>0$ My attempt $$\log a\le \frac ax-\frac1x$$ $$a\le \frac{\exp(a/x)}{\exp(1/x)}$$ But I ...
0
votes
1answer
37 views

Find all odd numbers $n$ such that $q= \frac{\ln(3n+1)}{6\ln(2)}$ is also an odd number. [on hold]

Find all odd numbers $n$ such that $$ q= \dfrac{\ln(3n+1)}{6\ln(2)}$$ is also an odd number.
1
vote
2answers
35 views

Why can we take the log of both sides?

I was watching a video that proves the "Log of a power" rule. I'm just having trouble understanding the loga(a^x) = x rule - which he uses in the proof And I don't get why you can log both sides. ...
0
votes
0answers
16 views

How to calculate anti log for this special case?

While solving a numerical , I came to a step where log k = 2.45 x 10^-3 So K = anti log ( 2.45 x 10^-3 ) But how to calculate the anti log of this number ? I just know to calculate the anti log ...
3
votes
1answer
28 views

Find derivative of tricky logarithmic functions

Find the derivative of $y=(x^{x+1})(x+1)^x$ So this is what I have, $$\ln y=\ln[(x^{x+1})(x+1)^x]$$ $$= \ln x^{x+1} + \ln(x+1)^x$$ $$\frac{1}{y}y' = (1)(\ln x) + (x+1)\frac{1}{x} + (1)(\ln(x+1)) + ...
0
votes
2answers
36 views

For which values of $m$, $f(x)=mx$ intersect the function $g(x)=\log x$?

For which values of $m$, the function, $f(x)=mx$ intersect the function, $g(x)=\log x$ I suppose that this problems reduce to the next form. Find for which values of m, exist solution for the ...
1
vote
0answers
38 views

Sum of 2 different irrational logarithms = Irrational?

I am having some problems proving that the following sum is irrational or rational: $\log_2(3)+\log_3(2)$ = irrational. This is all I've got for now: $\log_2(3)=\frac mn \iff 2^{\frac mn}=3 \iff ...
1
vote
0answers
13 views

Logarithm of an applied permutation

Say I have a cyclic permutation $P$, a known input $x$, and a known output $y$ such that $$y = P^a x$$ for some $a$. Is there a good way to search for $a$ (i.e. better than brute force)? Are some ...
2
votes
5answers
187 views

Textbook clarification: $\log = \ln$

Textbook reads: All logarithms are natural logarithms: $\log = \ln$. Does this mean $n\log(n) = n\ln(n)$?
1
vote
0answers
40 views

Trajectory With Air Resistance

For a video game, I am trying to calculate the angle needed for a projectile to hit coordinates x,y (both non-zero) with air resistance, i used equations from this site, and derived a function of y ...
0
votes
1answer
24 views

How to seperate out a variable from a log

I have: $$20\ln(1 + r/4) = \ln(4/3)$$ I'm trying to solve for $r$. Now if it was just $\ln(r/4)$, it would be easy: $\ln(r) - \ln(4)$, but in this case with a $1 + $ in front, I'm a little confused ...
-1
votes
0answers
14 views

Logarithms, principle logarithms in complex plane

How were these solutions reached? (a) $\log2 = \log|2| + i\mbox{Arg}2 +2\pi n i = \log2 + i2\pi n $ (b) $\log i = \log|i| + i\mbox{Arg}i + 2\pi n i = \frac{i\pi}{2} + i2\pi n $ (c) $\log(1+i) = ...
0
votes
2answers
53 views

Why is $y{(\log_a(x))} = \log_a{(x^y)}$?

Why is $y{(\log_a(x))} = \log_a{(x^y)}$? I feel like I'm missing something here. Sorry if I put the title wrong..
0
votes
0answers
18 views

Estimation with logarithm

I have to prove: $q$ an integer $\geq 2$, $\tau$ a real number $\geq 2$ and $\sigma \leq 1- \frac{1}{\log q\tau}$, $M\geq 0$. Then $M^{1-\sigma }\leq (q\tau )^{-\sigma }$ and $(M+1)^{-\sigma }\leq ...
1
vote
1answer
24 views

Tricky logarithm problem

I having a problem in this logarithm problem involving modulus- Solve for x |x-1|^((log(x))^2-2log(x))=|x-1|^3 Bases same so powers equal. If I take log x as a then I get the following quadratic- ...
0
votes
0answers
19 views

Logarithm inequality theoretical problem

I just want to ask why is it that the inequality sign reverses when we take antilog of an in equation. I understand it a bit practically but is there a proof? Like if log(f(x))< log(g(x)) Then ...
5
votes
2answers
435 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
0
votes
1answer
26 views

What is $log( b,a)$ according to google?

I expected that $log(b, a)$ represents $log_ba$. However this is not what google calculates for you if you type that into the search bar. For example, google says $log(4,2) \approx 0.62324929039$. ...
0
votes
2answers
35 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
2
votes
3answers
45 views

exact roots of $e^{ax}-x=0$

How can I find the general solution to (not a numerical approximation) $e^{ax}-x=0$ as a function of $a$. I thought maybe something like $\frac{ln(x)}{a}$.
2
votes
3answers
36 views

evaluating derivative of $\log_4(2x^2+1)$

Find the derivative and evaluate at $f\;'(2):$ $$\log_4(2x^2+1)$$ $\log_4(2x^2+1)=y$ $4^y=2x^2+1$ $4^y\ln4 \times y\;'=4x$ $y\;'=\dfrac{4x}{4^y\ln4}\implies \dfrac{4x}{(2x^2+1)\ln4}$ What ...
1
vote
4answers
30 views

Evaluating $\frac{d}{dx}\sqrt[4]{\ln(12-x^2)}$

Find Derivative and evaluate at $x=1$: $$ \frac{d}{dx}\sqrt[4]{\ln(12-x^2)} = (\ln u)^{1/4} $$ $$v=(v)^{1/4} \implies v=\ln\;u, v\;'=\dfrac{1}{u}(u\;')$$ $$y\;'=\frac{1}{4}v^{-3/4}\; \times ...
0
votes
2answers
36 views

How to estimate $\sum_{n=0}^\infty (\log\log2)^n/n! $ from below?

How to show that the following inequality holds: $$\sum_{n=0}^\infty \frac{(\log\log2)^n}{n!}>\frac 35$$ Is it possible to prove this using induction?
4
votes
5answers
129 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
0
votes
0answers
11 views

Exponential decay and logarithmic functions

How do you use experiential decay functions and logarithmic to create a mathematical model to compare the ages of two bones (Bone A and Bone B). When Bone A contains $3$ times the amount of ...
0
votes
3answers
41 views

Evaluating $\frac{\operatorname d \! \phantom x}{\operatorname d\!x}\frac{4}{\ln(x^2+2)}$

$\dfrac{\operatorname d \! \phantom x}{\operatorname d\!x}\dfrac{4}{\ln(x^2+2)}= \dfrac{4}{\ln u}$ $u=x^2+2$ $u\;'=2x$ $y\;'=\dfrac{4}{\dfrac{1}{u}} \times (u\;') \implies ...
0
votes
3answers
21 views

Derivative of $\frac{d}{dt}\ln(6t^2+9t+12)=$

$\dfrac{d}{dt}\ln(6t^2+9t+12)=$ $y=2\ln(6t)+\ln(9t)+\ln(12)$ $y\;'=2\dfrac{1}{6t}(6)+\dfrac{1}{9t}(9)+0$ $=\dfrac{12}{6t}+\dfrac{9}{9t}=\dfrac{2}{t}+\dfrac{1}{t}$ What am I doing wrong?
9
votes
1answer
144 views

Log integrals IV

It can be determined that the integral \begin{align} \int_{0}^{\pi/2} \frac{x}{\sin(x)} \ln\left(\frac{1+\cos(x) - \sin(x)}{1+\cos(x) + \sin(x)} \right) dx \end{align} has a finite value. Is there an ...
6
votes
1answer
103 views

Integral ${\large\int}_0^1\frac{\ln^2\ln\left(\frac1x\right)}{1+x+x^2}dx$

Gradshteyn & Ryzhik, 7th ed., p. 570, formula 4.325(5) give the following definite integral: ...
1
vote
4answers
55 views

How to evaluate $\lim_{x \to 0} ( \ln(1 - \sin x) + x)/x^2$ without using l'Hôpital?

How to evaluate $$\lim_{x \to 0} \frac{\ln(1 - \sin x) + x} {x^2}$$ without using l'Hôpital? I am not able to substitute the right infinitesimal. Is there a substitute? Background We have yet not ...
-2
votes
1answer
61 views

how to evaluate $ \lim_{x\to 0} \frac{\ln (x)}{1-x} $?

i know there is a special limit that is $$\lim_{x\to 0} \frac{\ln(x+1)}{x} = 1$$ but from here i cant determine the value of $$ \lim_{x\to 0} \frac{\ln (x)}{1-x} $$ could someone offer some kind ...
3
votes
3answers
57 views

Logarithmic inequality for a>1

Is $\log_{\sqrt a}(a+1)+\log_{a+1}\sqrt a\ge \sqrt6$ always true for $a>1$? What is the approach? Do we check the first a's and then form a induction hypothesis?
6
votes
2answers
93 views

How to Solve : $ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right) $

$ A =\frac{1}{6}\left((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3\right).$ Solve for $2^A.$ (no calculators or graphs are permitted) The way I went about solving this problem was using ...
3
votes
3answers
61 views

Approximation of Natural Logarithm using arithmetic.

A friend of mine posed this question to me a couple days ago and it's been bugging me ever since. He told me to take the square root of 5 twenty times, subtract 1 from it, and then multiply it by ...
0
votes
1answer
34 views

How to solve: $x^22^{x+1} + 2^{|x-3|+2} = x^22^{|x-3|+4} + 2^{x-1}$

Any help would be appreciated. :) I tried splitting the equation about $x=3$, but the terms $x^2$ and $2^x$ Together in the equation(s) are troubling me. I don't know why I'm unable to apply the ...
0
votes
2answers
45 views

Step by step (show-your-work) example on how to solve a log problem algebraically

It's been a long time since I have done calculus so if someone could please refresh my memory on the steps to solve the following problem algebraically that would be most appreciated. I am interested ...
0
votes
2answers
28 views

How to solve for $x$ when $a>0$: $log_{a+x}x \le log_ax^2 $ [closed]

After reaching $log_{a+x}x\le 2log_ax$ and concluding $x>0$ I'm not able to solve further. Hint?
2
votes
3answers
43 views

Is there an approximation to the natural log function at large values?

At small values close to $x=1$, you can use taylor expansion for $\ln x$: $$ \ln x = (x-1) - \frac{1}{2}(x-1)^2 + ....$$ Is there any valid expansion or approximation for large values (or at ...
-1
votes
2answers
41 views

solve $\log_3^2(x)-\log_2(x)=2$

The solution for the equation $\log_3^2(x)-\log_2(x)=2$ is a) $s=\{2,-1\}$; b) $s=\{6,-3\}$; c) $s=\{9, 1/3 \}$; d) $s=\{27, 1/9 \}$; e) $s=\{ 1/6 ,12\}$ It was given in a test at school and I ...