Questions related to real and complex logarithms.

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7
votes
2answers
111 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, ...
2
votes
3answers
16 views

problem solving logarithmic equation and reaching an equivalence

ok so i've had a problem trying to simplify the $\ln\left[ \sqrt{1+\frac{u^2}{a^2}} + \frac{u}{a} \right]$ and this is supposed to be equal to : $\ln [ \sqrt{a^2+u^2} + u ]$ how is this posible ?? ...
-1
votes
3answers
52 views

Integration by parts: $\int x\ln x^2 \,dx$

Problem: $\int x\ln x^2 \,dx$ So what I did first was make $u = \ln x^2$ and $dv = x$ Then I solved by getting the derivative of $u$ and the anti derivative of $dv$ and I got $du = 1/x^2 $ and $v = ...
3
votes
1answer
73 views

Finding $\int_{0}^{1} \frac{\log(1+x)}{1+x^2} {\rm d}x$ by differentiating under the integral sign.

I've tried to find this integral by the method already outlined in the title. I decided to let $$ \displaystyle I(\alpha) = \int_{0}^{1} \dfrac{\log(1+\alpha x)}{1+x^2} \text{ d}x. $$ From this ...
1
vote
1answer
29 views

Function related to Harmonic numbers, the Pascal triangle, Logarithmic integral and the Polylogarithm.

What function satisfies the following: Let the matrix: $$\displaystyle T = \left(\begin{matrix} 1&0&0&0&0&0&0&\cdots \\ 1&1&0&0&0&0&0 \\ ...
5
votes
2answers
84 views

Is $(\log(n))!$ a polynomially bounded function?

Is the following statement true? How would you prove it? i.e. Is it a polynomially bounded? $$ \lceil \lg(n) \rceil ! \in O(n^k) $$ How about $$ \lceil \lg \lg(n) \rceil ! \in O(n^k) $$ Thanks a ...
1
vote
2answers
47 views

Checking a possible logarithm identity: $(\sqrt{2})^{\lg n} \stackrel{?}{=} 2^{\sqrt{2\lg n}}$

I have to check if $(\sqrt{2})^{\lg n} = 2^{\sqrt{2\lg n}}$. My idea was to take logs: $\lg\ (\sqrt{2})^{\lg n} =\lg(2^{\sqrt{2\lg n}})$. But how to simplify further? What should I do next? Please, ...
-1
votes
1answer
13 views

iterated logarithm equation misunderstanding

I am trying to understand iterated logarithms. How anybody explain why $lg^{*}n = lg^{*}(lg\ n)$? What law can I apply to prove this equation?
1
vote
1answer
65 views

Solve $x\log x=y$

I have the following equation, $x\log x=y$. Is it possible to solve $x$ in terms of $y$. I think it is not possible but I am not sure.
1
vote
2answers
25 views

Prove that the binary representation of a number n will use floor(lg(n)) + 1 bits.

I'm taking Computer Algorithms class and one of my problems is from Skiena's Algorithm Design Manual, 2-41: Prove that the binary representation of $n \ge 1$ has $\lfloor \lg n \rfloor +1$ bits ...
0
votes
1answer
21 views

Logarithm Base Question

Suppose you have a integer n. Log2(n) is supposed to be ~ the number of times you have to divide n by 2 until you reach one. Now let's say you want to know ~ the number of times you have multiply n by ...
2
votes
0answers
27 views

Comparing asymptotic growth of logarithmic functions by reasoning

As an exercise, we're sorting functions according to their asymptotical growth. When comparing these two functions, I'm getting stuck: $n^2/(\log_2 n)^3$ versus $n \log_2 n$. Using limits I am ...
0
votes
0answers
27 views

Find the partial derivative with respect to y of the function $f(x,y)=ye^{xy}$

My solution was $e^{xy} + xy e^{xy}$, but when I checked the solution manual it said the answer is $xy e^{xy} \log e + e^{xy}$. So I solved each function for $y$ by setting them each equal to $0$. ...
2
votes
1answer
64 views

Why does $\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}=\log_{\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}}(x)?$ (Error)

I might be being very silly here, but I can't for the life of me see why $$\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}=\log_{\sqrt{x-x^{\frac{1}{x}^{\frac{1}{x}}}}}(x)$$for $x\in \mathbb{Z}, x>1$? ...
0
votes
1answer
35 views

Graphing natural logarithms

I don't know how to obtain the graph of these functions. Could someone please help me? I know what the graph of ln looks like, but other then that I don't know where to go. Thank you for any help ...
0
votes
2answers
36 views

Inequalities with logarithms and limits

For my analysis homework, I am to show that $\lim_{n \to \infty} \frac{3^n}{n!} = 0$ using the epsilon definition. My approach is to invoke the squeeze theorem and show that the above sequence is less ...
3
votes
1answer
32 views

Growth rate of logarithmic function?

Just curious about the growth rate of the logarithmic function: Does there exist a real number $n$ such that $lim_{x \to \infty} \frac{(ln(x))^{n}}{x}$ diverges (does not converge to $0$)? Thanks in ...
0
votes
1answer
36 views

help with logarithmic integration.

I've been googling some tutorials on integrating logarithms for my calc 2 class and I've found a lot of good stuff. Unfortunately nothing has answered how to handle a problem that I have. I've tried ...
0
votes
0answers
45 views

How is $O(\log(\log(n)))$ also $O( \log n)$?

How is $O(\log(\log(n)))$ also $O( \log n)$? I have seen this result somewhere with this but I still didn't quite understand how this is true. This would also help me compute Big Omega of the ...
1
vote
2answers
57 views

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal?

Algebraically, how are $-\ln|\csc x + \cot x| +C $ and $\ln| \csc x - \cot x|+C$ equal? I know both of these are the answer to $\int \csc x \space dx$, and I am able to work them out with calculus ...
-1
votes
3answers
26 views

Solve for two variables, two equations with exponents [closed]

Solve for both k and x, where $5=k(300)^x$ and $80=k(600)^x$
0
votes
1answer
27 views

Solving $1/n^{\lg (n)}$

I am struggling with logarithms and their computation when it comes to computing time complexity. I have a simple complexity: $\frac{1}{n^{\lg (n)}}$, where the logarithm base is 2. How can I reduce ...
12
votes
1answer
303 views

Evaluating $\int_0^{\Large\frac{\pi}{2}}\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan(x)}\right)^3dx$

Using the method shown here, I have found the following closed form. $$ \int_0^{\!\Large \frac{\pi}{2}}\!\!\left(\frac{1}{\log(\tan x)}+\frac{1}{1-\tan x}\right)^2\! \mathrm dx= ...
0
votes
0answers
18 views

Exponential and Logarithmic Differentiation.

Q. If $xe^{xy}=y+sin^2x$, then find $\frac{dy}{dx}$ at x=0. If we differentiate the function directly as follows: $e^{xy}+xe^{xy}\left[y+x\frac{dy}{dx}\right]=\frac{dy}{dx}+sin\left(2x\right)$ At ...
2
votes
2answers
54 views

Showing if $n \ge 2c\log(c)$ then $n\ge c\log(n)$

Is this true that if $n \ge 2c\log(c)$ then $n\ge c\log(n)$, for any constant $c>0$? Here $n$ is a positive integer.
35
votes
2answers
916 views

Closed form for ${\large\int}_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx$

Here is another integral I'm trying to evaluate: $$I=\int_0^1\frac{\ln(1-x)\,\ln(1+x)\,\ln(1+2x)}{1+2x}dx.\tag1$$ A numeric approximation is: ...
3
votes
2answers
84 views

Why does $\lim_{x\rightarrow\infty} x-x^{\frac{1}{x}^{\frac{1}{x}}}-\log^2x=0?$

Why does $$\lim_{x\rightarrow\infty} x-x^{\frac{1}{x}^{\frac{1}{x}}}-\log^2x=0?$$ Moreover, why is $$x-x^{\frac{1}{x}^{\frac{1}{x}}}\approx\log^2 x?$$
1
vote
0answers
58 views

Solving ${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x \le c_2$

Is there any way to solve $${c_1}^x+\sqrt{\frac{\log(x)x}{2}}+3\log(x)x\le c_2,$$ for $x>1$, $0<c_1<1$, and $0<c_2<<1$? Thanks
5
votes
2answers
326 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
0
votes
1answer
79 views

How to find numbers $k$ such that $kx - \ln(ex + 1-x) $ is positive on $(0,1]$?

I want to find a condition on $k$ such that $g(x)= kx - \ln(ex + 1-x) > 0$, $x\in [0,1] $. At zero the function is zero. So, to find a condition on $k$ I use $g'(x) > 0$ i.e. $$ k > ...
0
votes
1answer
17 views

Distributing out log equation

$$\log_{27}x = 1 - \log_{27}(x-0.4)$$ $$\log_{27}(x(x-0.4))=1$$ $$x=5.4,\, x=-5$$ I'm confused on the second line. How come it is not $\log_{27}(x+x-0.4)$?
4
votes
1answer
60 views

Simplify an iterated function

If we iterate the function $f(x) = \ln(x + 1)$, we get: $$f(f(x)) = f^2(x) = \ln(\ln(x + 1) + 1)$$ $$f(f(f(x))) = f^3(x) = \ln(\ln(\ln(x + 1) + 1) + 1)$$ $$f(f(f(f(x)))) = f^4(x) = \ln(\ln(\ln(\ln(x + ...
0
votes
2answers
23 views

Simple Logarithmic Question

I have the following equation: $\log(S_n) = \log(u)[2T-n]\,\,$ I was just wondering how $S_n = u^{2T-n}$ is then obtained? Thank You
2
votes
4answers
91 views

Is there any expansion for $\log(1+x)$ when $x\gt 1$?

Is there any expansion for $\log(1+x)$ when $x\gt 1$ ?
0
votes
0answers
44 views

Would the growth rate for base 2 and 10 logs be the same?

Since $\log_{2}(x) = \frac{\log_{10}(x)}{log_{10}(2)}$ and $\log_{10}(2)$ is just a constant, would their growth rate be the same?
0
votes
1answer
18 views

Bounding a logarithmic relation

If I have the following relation $T(n) \le an\lceil \operatorname{lg} (n) \rceil - an +2bn + n$, is it possible to bound $T(n)$ such that it is in the form $T(n) \le an\operatorname{lg}(n) + bn $ for ...
1
vote
2answers
33 views

What log rule was used to simply this expression?

I'm unclear how the left side is equal to the right side. $$365\log(365) - 365 - 305\log(305) + 305 - 60\log(365) = 305\log\left(\frac{365}{305}\right)-60$$ I know $\log(a) - \log(b) = \log (a/b)$ ...
2
votes
1answer
33 views

Having trouble understanding why the $r$-th mean tends to the geometric mean as $r$ tends to zero

I am having trouble understanding the proof of Theorem 3 in "Inequalities" by Hardy, Littlewood and Pólya. This theorem states that the $r$-th mean approaches the geometric mean as $r$ approaches ...
0
votes
1answer
20 views

Domain of a multiple logarithmic function.

Find the domain of the following function: $f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$ My text provides a solution which goes like: => ...
0
votes
2answers
62 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
0
votes
1answer
51 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
3
votes
3answers
277 views

Find a real entire function $f(z)$ asymptotic to $\ln(x^2+1)$ for real $x$.

Find a real entire function $f(z)$ asymptotic to $\ln(x^2 +1)$ for real $x$. More specific I want $f(0)=0$ and $\frac{1}{2} \ln(x^2+1) < f(x) < 2 \ln(x^2+1)$. Or prove it does not exist.
2
votes
2answers
92 views

Calculating the integral of a logarithmic expression.

The problem I have been working with is $$\int \frac 1{\sqrt x(1+\sqrt x)}\,dx$$ The first step I did to solve this question was to set $u= 1+ \sqrt x$ the set $du = (1/2) x^{-1/2}$ Then I set ...
3
votes
0answers
64 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
19
votes
1answer
286 views

Integral ${\large\int}_0^1\ln(1-x)\ln(1+x)\ln^2x\,dx$

This problem was posted at I&S a week ago, and no attempts to solve it have been posted there yet. It looks very alluring, so I decided to repost it here: Prove: ...
4
votes
2answers
777 views

Solving a logarithmic expression without a calculator

How do I find the value of this logarithmic expression without using a calculator? I'm trying to relearn algebra, but this problem has me scratching my head, and my Google tutorial searches are ...
3
votes
1answer
53 views

How to get 2 using a standard scientific calculator without pressing the number buttons 0 to 9 and the buttons $+-\times\div$?

I was challenged by a friend to get a number 2 by using a standard scientific calculator but without pressing the number buttons 0 to 9 and the buttons $+-\times\div$. I could get 1 from $\ln e=1$. ...
1
vote
2answers
90 views

solve for $x$ without using softwares $\log_{\sqrt{x}}2+\log_6x^x=4$

Is there any nice way to solve this equation without wolfram? $\log_{\sqrt{x}}2+\log_6x^x=4$ Thanks.
0
votes
1answer
19 views

How do you solve a recurrence with a functin through induction?

I found the answer in part-A by substitution, as O(n) from; T(n/2^k) = T(1).... n/2^k = 1..... so k = 1og2(n)..... T(log2(n)) = T(n/n)+5.... so O(n) IS THE ANSWER, Correct me if am wrong because am ...
8
votes
2answers
183 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about ...