Questions related to real and complex logarithms.

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3answers
49 views

Proof of the fact that ln(a) = f '(0) for f(x) = a^x?

Looking over notes from class today and wanted to know if there is any type of proof for the fact that $\ln(a) = \lim_{h\to0}(a^h-1)/h$, which is just $f '(0)$ for any function of the form $f(x) = ...
8
votes
1answer
149 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
3
votes
1answer
195 views

Calculus integral evaluation using substitution

I have to find this integral: Evaluate the integral using an appropriate substitution $$\int\dfrac{8e^x+7e^{-x}}{8e^x-7e^{-x}}\mathrm dx.$$ I've tried my solution $\ln\Big[15\cdot \sinh(x) + ...
2
votes
2answers
33 views

Logarithmic function with strange bases

Given $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n$. I have rewritten $\log_{3n} 45$ as $\dfrac{\log_{4n}45}{\log_{4n}3n}$ and multiplied to get $\log_{4n} 40\sqrt{3}\cdot\log_{4n}3n = \log_{4n} ...
1
vote
2answers
29 views

Stuck with finding the domain of a function with a logarithm

Find the domain of the function $$g(x)=\log_3(x^2-1)$$ This is what I got so far: $$\{ x\mid x^2-1>0\} =$$ $$\{ x\mid x^2>1\} =$$ $$\{ x\mid x>\sqrt { 1 } \}= $$ I don't know where to ...
1
vote
2answers
49 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
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1answer
24 views

Taking the log of both sides to determine big Theta/Omega/O

I've managed to confuse myself over this detail: Obviously: $n^2 \notin \Theta(n)$ Now if we take the $\log$ of both sides, we get: $$\log(n^2) \leq \log(cn)$$ $$2\log(n) \leq \log(c) + \log(n)$$ ...
1
vote
1answer
38 views

Principal square root of a product of complex numbers with positive real part

Given $n$ complex numbers $z_i$ with $\Re z_i>0$, why is it that $$\prod_i\sqrt{z_i}=\sqrt{\prod_i z_i}?$$Numerically, this appears to be the case, however, I don't see an easy way to prove it.
0
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3answers
82 views

How to solve $2\ln(x) = \sqrt{x}$ ? ln = natural logarithm

I used Microsoft Mathematics and it says $x$ is approximately $2.04\dots$ but, how do you prove it? Edit: I'm sorry if I wasn't clear enough with my question. I don't want to prove that two roots ...
0
votes
1answer
25 views

Differential equation $xy'+2y=0$ and the form of arbitrary constant in its general solution

If I'm solving the differential equation in the title I will get to: $$\log(y)=-2\log(x)+c$$ then I'll get $y=e^c/x^2$ eith arbitrary constant $c$. So I know I can write $y=d/x^2$ where $d$ is an ...
11
votes
0answers
137 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
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0answers
36 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
39 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
63 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
23 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
53 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
41 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
30 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
75 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
135 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
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1answer
33 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
25 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
57 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
28 views

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

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 $9log_x5$ into $log_x5^9$. Can anyone guide me onto the right path to solve this?
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.
1
vote
1answer
31 views

Lambert- W -Function calculation?

I have an equation of the form nlogn=x . Upon searching , i came across the term "Lambert- W -Function",but couldn't find proper method for evauation, and ...
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
42 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
63 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
30 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
58 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
26 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
12 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 ...
10
votes
3answers
302 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
22 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
67 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
54 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
58 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 ...
5
votes
2answers
76 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
423 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? ...