Questions related to real and complex logarithms.

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2
votes
1answer
20 views

Simplification of an expression containing $\operatorname{Li}_3(x)$ terms

In my computations I ended up with this result: $$\mathcal{K}=78\operatorname{Li}_3\left(\frac13\right)+15\operatorname{Li}_3\left(\frac23\right)-64\operatorname{Li}_3\left(\frac15\right)-102 ...
2
votes
1answer
26 views

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$ I stucked at the denominator.
0
votes
1answer
20 views

Solving an equation with exponents by using logarithms

Solve the equation $$0.25^5 = 4^{(5x-3)/3} \cdot (0.125)^{6x}$$ So would I just bring down the exponents by taking the log of each constant?
0
votes
1answer
22 views

Log arithmic Equation - Graph curved line

I'm recreating the graph picture below with equations. Using the online graphing tool "Desmos": These are all the equations I have done so far, with there restrictions top stop at specific points. ...
7
votes
0answers
51 views

Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
0
votes
3answers
51 views

Logs and indices questions

Hi can anyone solve these two questions using logs and indices a. $$4^{2x}-2^{x+1}=48$$ b. $$6^{2x+1}-17*{6^x}+12=0$$ Thanks.
0
votes
1answer
42 views

Graphing picture equations - Curve Lines

I'm basically trying to recreate the graph picture below. Using a online graphing tool "Desmos": I managed to create the equations for the straight lines and circles for the sunset picture. ...
1
vote
3answers
36 views

Logarithmic inequality

Solve the inequality: $$ \log_8(x^2-4x+3) < 1 $$ $$ \log_8(x^2-4x+3) < \log_8(8) $$ $$ \log_8(x^2-4x+3) - \log_8(8) <0 $$ $$ \log_8 [(x^2-4x+3)/8] < 0 $$ Thats what I did for the question ...
0
votes
2answers
17 views

Question on solving a Logarithmic equation

$\ln(x+3)^{\frac{1}{2}} + \ln (4x-3)^{\frac{1}{2}} = \ln (5)$ So I understand that in order to solve this log function, I would have to square the square roots to simplify the equation. But how ...
4
votes
2answers
54 views

Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
9
votes
0answers
80 views

Closed form for ${\large\int}_0^1\frac{\ln^{\color{magenta}3}x}{\sqrt{x^2-x+1}}dx$

This is a follow-up to my earlier question Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$. Is there a closed form for this integral? ...
0
votes
1answer
19 views

Arithmetic mean using logarithm

I have logarithm data in dBm and I want to compute the arithmetic mean of this data. The problem is, I'm not sure if I can directly compute the mean using logarithm, adding and then dividing for the ...
0
votes
2answers
22 views

Substituting logs

If $b=log_3(x),$ what value of $x$ satisfies $log_b(log_3(x^2))=3?$ I just started learning this topic by myself. I wanted to know if my working is correct. If not can someone help me with this ...
0
votes
1answer
37 views

A^N - B^N = C, A,B,C are known, solve for N

As title says: $$A^N - B^N = C,$$ $A,B,C$ are known, solve for $N$. This is substracted from a bigger formula where this N is one of the parameters to be calculated. I have tried it with: ...
2
votes
2answers
59 views

Simplify $\sinh (\log (x))$

$$\sinh (\log (x))=\frac{x^2-1}{2 x}$$ However I do not see how this is done, here is an idea I had but I'm probably way off: $$\sinh \left(\ln \left(\frac{1}{2} ...
0
votes
1answer
23 views

Comparing the order of convergence $\mathcal{O}( h^2 |\log(h)|)$

I don't have any intuition in judging how fast a term of the order $\mathcal{O}( h^2 |\log(h)|)$ is decreasing as $h \to 0$, so i tried comparing it with terms of the form $\mathcal{O}( h^\alpha )$ ...
0
votes
1answer
32 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 ...
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 ...
7
votes
2answers
95 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 ?? ...
-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
1answer
60 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
votes
0answers
29 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
19 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
68 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
46 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
60 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
17 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 $0$. ...
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 $x\in \mathbb{Z}, x>1$? ...
-1
votes
0answers
23 views

Producing a log table base 10 [closed]

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 ...
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
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 ...
0
votes
1answer
34 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
39 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
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 ...
-1
votes
3answers
24 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
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 ...
11
votes
1answer
141 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
13 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
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.
29
votes
1answer
618 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: ...
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 $$x-x^{\frac{1}{x}^{\frac{1}{x}}}\approx\log^2 x?$$
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
308 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
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 > ...