Questions related to real and complex logarithms.

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3
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6answers
63 views

Prove that $\lim_{x \to \infty} \frac{\log(1+e^x)}{x} = 1$

Show that $$\lim_{x \to \infty} \frac{\log(1 + e^x)}{x} = 1$$ How do I prove this? Or how do we get this result? Here $\log$ is the natural logarithm.
1
vote
0answers
16 views

Interpretation of difference log points in a regression

The post "How to interpret the difference in log points" shows how to interpret differences in log values still in log form. As an extension to this, however, I would like to know how to consider an ...
0
votes
1answer
18 views

What is the complexity of halving the size of an $n$-bit number every time.

I was discussing this question with my fiend the other day and was hoping to get some confirmation from someone if the logic I used is correct. Suppose that we have a number $N$ in base 2 ie ...
1
vote
3answers
160 views

What is the best way to calculate log without a calculator?

As the title states, I need to be able to calculate logs on paper without a calculator. For example, how would I calculate $log(25)$ ?
0
votes
1answer
9 views

Different domains for (apparently) equivalent functions

Let's look at: $f_1(x)=ln(x^2-4)$ $f_2(x)=ln(x-2)+ln(x+2)$ Every high school student can tell they are the same, but the first is defined only for $\{x<-2\}\cup\{x>2\}$, and the latter is ...
6
votes
2answers
38 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 ...
4
votes
4answers
10k views

Units of a log of a physical quantity

So I have never actually found a good answer or even a good resource which discusses this so I appeal to experts here at stack exchange because this problem came up again today. What happens to the ...
0
votes
1answer
17 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
32 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
23 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
2answers
75 views

Lambert- W -Function calculation?

I have an equation of the form: $$ n \log n = x $$ Upon searching I came across the term "Lambert- W -Function" but couldn't find a proper method for evaluation, and neither any computer code for ...
6
votes
5answers
535 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
61 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
21 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
81 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 ...
2
votes
4answers
39 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
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
44 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 ...
0
votes
1answer
38 views

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

Find all odd numbers $n$ such that $$ q= \dfrac{\ln(3n+1)}{6\ln(2)}$$ is also an odd number.
1
vote
5answers
181 views

How can I calculate $\lim_{x \to 0} \log(\cos(x))/\log(\cos(3x))$ without l'Hopital?

How can I calculate the following limit without using, as Wolfram Alpha does, without using l'Hôpital? $$ \lim_{x\to 0}\frac{\log\cos x}{\log\cos 3x} $$
1
vote
2answers
52 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
2answers
38 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 ...
0
votes
2answers
55 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..
3
votes
1answer
34 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)) + ...
1
vote
0answers
41 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 ...
2
votes
5answers
192 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
16 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 ...
1
vote
0answers
43 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
26 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 ...
4
votes
7answers
301 views

Study convergence of the series of the $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sin{(\ln{n})}}{n^a}$

Question: Study convergence of the series $$\sum_{n=1}^{\infty}(-1)^{n-1}\dfrac{\sin{(\ln{n})}}{n^a},a\in R$$ My try: Thank you DonAntonio help, (1):$a>1$, since $$|\sin{(\ln{n})}|\le1$$ so ...
0
votes
0answers
20 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 ...
5
votes
2answers
458 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?
1
vote
1answer
39 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- ...
1
vote
4answers
58 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 ...
0
votes
0answers
20 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 ...
1
vote
1answer
72 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}) ...
0
votes
1answer
29 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
38 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 ...
1
vote
4answers
32 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 ...
1
vote
2answers
95 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
vote
3answers
244 views

Derivative of $(\ln x)^{\ln x}$

How can I differentiate the following function? $$f(x)=(\ln x)^{\ln x}.$$ Is it a composition of functions? And if so, which functions? Thank you.
3
votes
3answers
159 views

$x \sim y \implies \log x \sim \log y$?

Does $x \sim y \implies \ln x \sim \ln y$? I would have thought not, but the following has almost persuaded me otherwise: Assume $x \sim y.$ Does this imply that $$\tag{1}I = ...
2
votes
3answers
37 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
3answers
46 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 $\frac{ln(x)}{a}$.
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?
5
votes
5answers
138 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 ...
5
votes
1answer
84 views

Log Log Integrals II

The integral \begin{align} I_{4} = \int_{0}^{1} \ln(1-x) \ \ln^{2}\left( \ln\left(\frac{1}{x}\right) \right) \ \frac{dx}{x} \end{align} can be expressed as \begin{align} I_{4} = \zeta^{''}(2) - ...
9
votes
1answer
190 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 ...
0
votes
3answers
42 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?