Questions related to real and complex logarithms.

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0
votes
1answer
23 views

General formula for a series

I am trying to solve series of the form, T(n) = T(n/4) + clog(n) I am able to formulate a general formula for the T(n) term for the nth term. Its of the form ...
2
votes
2answers
53 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.
0
votes
1answer
24 views

Logarithm multiplication property error, can't figure out why.

I know there is a mistake and where it is but I can't figure out why. Equation: $$ 3+2(12^{x+1}) = 291 $$ From here I do: $$ 2(12^{x+1}) = 291-3\\ 2(12^{x+1}) = 288\\ $$ Then I take the natural ...
6
votes
2answers
83 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
15 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 ?? ...
3
votes
1answer
56 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 ...
3
votes
3answers
235 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.
-1
votes
3answers
47 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
4answers
1k views

Is it possible to solve logarithm when base is unknown?

Is it possible to solve logarithm equation when the base of the logarithm is unknown but the result is known. Here is an example: $$ \log_{X} (\frac{223}{150}) = 20 $$ This basically means that if x ...
-1
votes
0answers
28 views

How can I count logarithm of sinus?

I've red a book when was the information about logarithm of sinus/cosine etc. There was description of the logarithm sinus/cosine tables and how to using that. But there wasn't description how to ...
1
vote
1answer
18 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
66 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
45 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
12 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
59 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.
18
votes
0answers
295 views
+200

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: ...
1
vote
2answers
16 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
18 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
11 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
24 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 $0$. ...
7
votes
2answers
2k views

Value of Summation of $\log(n)$

Context: I am learning Dijstra's Algorithm to find shortest path to any node, given the start node. Here, we can use Fibonnacci Heap as Priority Queue. Following is few lines of algorithm: ...
2
votes
1answer
62 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 $x\in \mathbb{Z}, x>1$? ...
5
votes
4answers
128 views

Is $\ln\sqrt{2}$ irrational?

I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which ...
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 ...
-1
votes
0answers
23 views

Producing a log table base 10 [on hold]

i have an assignment which says: Asume that a log table with the base 10 will be produced log10 for x=1+iδ for 0< The method used will be the "Latin method", the one where you decide the ...
7
votes
5answers
381 views

Hints on calculating the integral $\int_0^1\frac{x^{19}-1}{\ln x}\,dx$

I would be happy to get some hints on the following integral: $$ \int_0^1\frac{x^{19}-1}{\ln x}\,dx $$
1
vote
4answers
128 views

Solving the logarithimic inequality $\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$

I tried solving the logarithmic inequality: $$\log_2\frac{x}{2} + \frac{\log_2x^2}{\log_2\frac{2}{x} } \leq 1$$ several times but keeping getting wrong answers.
10
votes
1answer
131 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
2answers
35 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
29 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 ...
2
votes
4answers
90 views
0
votes
1answer
33 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 ...
1
vote
2answers
53 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 ...
0
votes
0answers
38 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 ...
0
votes
2answers
293 views

Find Log equation from data points

I have the following data points, (left hand column goes from 0-127, right hand column goes from 30-22000 hz. Is there any calculator I can use to find a "log" function of this data, so that it comes ...
-1
votes
3answers
24 views
0
votes
1answer
25 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 ...
0
votes
0answers
12 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 ...
0
votes
2answers
29 views

How to express the given quantity as a single logarithm? [closed]

I'm working some homework and I've hit a brick wall. How do I solve this? Express the given quantity as a single logarithm. $$\frac19\ln[(x+2)^9]+\frac12\ln(x) - \ln[(x^2+3x+2)^2]$$
3
votes
1answer
45 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
0answers
57 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
6
votes
4answers
296 views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
4
votes
4answers
370 views

log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$. So I was wondering where I have gone ...
3
votes
2answers
82 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 $$x-x^{\frac{1}{x}^{\frac{1}{x}}}\approx\log^2 x?$$
0
votes
1answer
78 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 > ...
5
votes
2answers
304 views

A logarithmic integral

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
2
votes
1answer
62 views

How does $\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2\;?$

So as the title says it all: How does $\;\left(\log \sqrt x\right)^2 = \frac 14(\log x)^2 \;?$ To be specific, why the removal of root, and how do we get 4 in denominator?
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
42 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
22 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