Questions related to real and complex logarithms.

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12
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0answers
143 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
1
vote
3answers
115 views

Solving $\log(x) = x-1$?

One can use Taylor series of the log or exp function to get the result that $x = 1$. I was wondering if there is any other simple solutions. Thanks a lot!
0
votes
0answers
38 views

Limit with logarithm and cos function.

limit(x tends to x/2)(cos.log tanx). Can anyone give a way? The problem is that i am ending with limit(h tends to 0)(cos.log 0). And as log 0 is undefined so i cannot do it anymore
2
votes
3answers
43 views

Logarithmic Differentiation - when to use?

Sorry if this is an ignorant or uninformed question, but I would like to know when I can (or should use) logarithmic differentiation. I haven't taken calculus in a while so I'm quite rusty. So, let's ...
2
votes
1answer
64 views

Number of digits in $12^{300}$

Given: $\log_{10}2= 0.3010$ and $\log_{10}3=0.4771 $, find the numer of digits in $12^{300}$ Options: $324,323,325,\text{Other}$ Actually I tried breaking 12 into 2*2*4.. And then tried to guess ...
1
vote
1answer
33 views

Different answer when simplifying before integrating

I have been trying to get my head around this for some time now... I solve the same integral in two ways but get two different solutions. Since there can't (surely) be any sort of ambiguity when ...
0
votes
2answers
38 views

How to determine the value of a variable in a equation with powers

I'm completely rusty on this How would be the way of determing the value of x in something like this $\ 100 = \frac{50}{(1 + x)^a} + \frac{50}{(1 + x)^b} + \frac{50}{(1 + x)^c}$ a, b, c are known ...
0
votes
2answers
24 views

proof - proving a proposition involving logarithms is true or false

I'm looking at my textbook and I'm not sure how to solve this to prove whether it's true or not. (there exists x in the real)(3^x = x^2 ) Any help would be good. Thank you.
1
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2answers
54 views

How to find all the intersection points of the two functions $\log(x!)$ and $x$?

I am trying to find where $\log(x!)$ and $x$ intersect, and am unable to do so rigorously. I eventually have $2^x = x!$, but I am unsure how to proceed from here. Any input as to how to go about ...
0
votes
4answers
90 views

Solve $\ln(x)+\ln(x-1)=0$ for $x$

Solve the following equation for x; $$\ln(x)+\ln(x-1)=0$$ What I did is the following but I'm pretty sure its wrong.. $$\ln(x)+\ln(x-1)=0$$ $$\ln(x)=-\ln(x-1)$$ $$e^{\ln(x)}=e^{-\ln(x-1))}$$ ...
1
vote
1answer
43 views

Series involving a Logarithm

Consider the series \begin{align} \sum_{n=1}^{\infty} \left[ \frac{n}{a} \ln\left(1 + \frac{a}{n}\right) - 1 + \frac{a}{2n} \right]. \end{align} Is there a closed form solution to this series and what ...
2
votes
2answers
31 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
0
votes
2answers
82 views

How did Newton calculate 3x7 by logarithm?

This is a story about Newton I read once when I was a child. Now that book is lost and I can only tell you what I remember. When Newton was young, he had been already famous in curiosity and ...
2
votes
2answers
137 views

How to solve this logarithm equation

$$\log_2\left\{\log_3\left[\log_4\left(x^{3x}\right)\right]\right\} = 0$$ How would I go about solving this? I tried doing $\log_4(x^{3x}))=0$ but I don't know how to incorporate the other logs
-1
votes
1answer
34 views

Calculate log of number less than raised to power

I want to calculate the value of 0.9 raised to power 17.I am using the log method. 17 * log(0.9).Am I doing this correctly?
1
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0answers
26 views

What does Heron's Algorithm have to do with the construction of logarithmic tables

i need a little help answering this question, what does Heron's Algorithm have to do with the construction of logarithmic tables. I know that Heron's algorithm is used for finding square roots, but ...
1
vote
1answer
33 views

Show $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$

I am trying to show that $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$ but I seem to get a conflicting result. What i did is: $n!=1*2*3*...*n \leq n*n*n*...*n=n^n$, so $\frac{n}{2} \log(n!) \leq ...
1
vote
3answers
59 views

What algorithm solves this problem? Non-linear measuring tape

A measuring tape is marked at 0, 5, 15 and 40. The distances between each mark are marked on top. At what distances should I mark 1 through 4, as well as 6-14 and 16-39? My math knowledge does not ...
1
vote
5answers
41 views

Find the range of values that $x$ can take if $9 \log_x5 = \log_5x$

I'm stuck on a homework question about logarithms. I can't work out how to do it, and all I've managed to do is turn $9 \log_x5$ into $ \log_x5^9$. Can anyone guide me onto the right path to solve ...
0
votes
3answers
69 views

How can I know $\int_1^x\frac{dt}{t}$ is the inverse of exponential function?

How can I know $\int_1^x\frac{dt}{t} \forall x>0$ is the inverse of exponential function assuming I've never heared of the natural logarithm.
2
votes
2answers
69 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 ...
0
votes
1answer
32 views

Proving $\lg n!=\Omega(n\lg n)$

In the answer given in the book for the proof of $\lg n=\Omega(n\lg n)$ there are several steps which I don't understand . $$\lg n!=\lg n+\lg(n-1)+\lg(n-2)+ ....+\lg(2)+\lg 1$$ Then it says that ...
0
votes
1answer
43 views

Following flash, a camera's battery begins to recharge the flash’s capacitor, which stores electric charge given by $Q(t) = Q_0(1 − e^{−t/a})$ [closed]

(The maximum charge capacity is $Q_0$ and $t$ is measured in seconds). (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of ...
1
vote
2answers
91 views

Summation of series involving logarithm: $\sum (n+2)\ln 2^n$

The following question is: Show that $\sum\limits_{r = 1}^n {r(r + 2)} ={n \over 6}(n+1)(2n+7).$ Using this results, or otherwise, find, in terms of $n$, the sum of the series ...
1
vote
1answer
31 views

Simplifying / Solving for $x$

I'm new here, looking for some help please. I've been at this question for 4+ hours, not getting anywhere, haha. $\log_2 (kx) = a$ Question asks to solve for $x$ So far my best try is $\log_2 ...
1
vote
3answers
66 views

Unable to differentiate $\cos(x) \cos(2x) \cos(3x)$ and $\sqrt{\frac{(x-1)(x-2)}{(x-3)(x-4)(x-5)}}$

I apologize for the lack of LaTeX. I will update this question with the proper LaTeX as soon as possible. I am having trouble with two differentiation exercise questions and was hoping someone could ...
1
vote
1answer
30 views

Principal branch of the complex logarithm does not always satisfy the product formula

My book asks to prove: $\text{Ln}[i \cdot (-1+i)]$ does not equal to $\text{Ln}(i) + \text{Ln}(-1+i)$ where $\text{Ln}$ gives the principal log of the complex number. I don't see why this is true ...
0
votes
4answers
43 views

Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...
0
votes
1answer
13 views

If the following numbers are put in order from smallest to largest then which of the numbers will be the middle number on the list?

If the following numbers are put in order from smallest to largest then which of the numbers will be the middle number on the list? A. $4\log(3)$ B. $0.5\log(144)$ C. $\log(4)+\log(5)$ D. ...
0
votes
3answers
40 views

If $f(x)=\frac{2^{2x}+2^{-x}}{2^{x}-2^{-x}}$ then evaluate $f(\log_2(3))$

If $$f(x)=\frac{2^{2x}+2^{-x}}{2^{x}-2^{-x}}$$ Then evaluate $f(\log_2(3))$. Can someone help me to understand the calculation? I figured out that the result is $7/2$ but I have problems solving ...
11
votes
3answers
325 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
3answers
37 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.
1
vote
1answer
23 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
97 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. ...
0
votes
3answers
55 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
67 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
41 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
19 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 ...
7
votes
2answers
87 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: ...
16
votes
1answer
434 views

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

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
20 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
25 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
47 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
62 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
25 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
34 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
27 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
108 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
51 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 = ...