Questions related to real and complex logarithms.

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0
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0answers
24 views

How to solve inequalities where the $x$ term appears inside the argument of multiple different functions?

We're asked to study the sign of the following function: $$\frac{x(\ln{x}+1)^2 - 2(\ln{x}+1)^2 - \frac{4}{x(\ln{x}+1)}}{(x(\ln{x}-1)^2)^2} \geq0,$$ in which the $x$ variable appears both outside ...
1
vote
2answers
48 views

Calculate Ln$(i^i)$

Calculate Ln$(i^i)$ My attempt: Ln$(z)$=$\ln|z|+i\arg z$ $$z=0+i^i=0+i\cdot i$$ $$|z|=\sqrt{0^2+i^2}=i\\ \arg z=\arctan(i/0)$$ $1.$ how it can be that the modulus equal to $i$? $2.$ how ...
1
vote
1answer
29 views

How to solve for a variable in logarithms

How do I solve this for $y$? $$u= 1 - \exp\left\{-\left(\frac{y-\theta}{\alpha}\right)^\gamma\right\}.$$ If I take the $\log$ I end up with $$\log(1-u) = ...
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0answers
38 views

Integral with logarithmic residue (Residue general method)

I was reading about this post, about Integration in complex plane with logarithmic residue: Integral with logarithm - residue And I thought about: what if $R(x)$ function is inside the logarithm? ...
1
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0answers
47 views

Defining a continuous complex logarithm on open set $U \subset \mathbb{C}$

Suppose you are given an open set $U \subset \mathbb{C}$ and a continuous function $f: U \rightarrow \mathbb{C}-\{0\}$. And $f$ has the next property: For every closed loop $ c: I \rightarrow U$ ...
0
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2answers
52 views

Prove $\log(x)^{10} < x$ (for $x > 10^{10}$)

I need to prove that $\log(x)^{10} < x$ for $\ x>10^{10}$ It's pretty clearly true to me, but I need a good proof of it. I tried induction, and got stuck there.
1
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2answers
45 views

How to solve this equation using logs

How do solve this equation for x using logarithms? $$4^x = 6^x-3$$ If it is not possible using logarithms, please provide another way. Thank you in advance
3
votes
1answer
31 views

Summation of $\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$ and $\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any progress. ...
0
votes
0answers
24 views

How to expand in terms of inverse logarithms?

I'm currently working with a matched asymptotic expansion problem. Currently, I have a function $f$ that can be expanded as: $$f = f_0 + \frac{f_1}{\ln{\epsilon}} + \frac{f_2}{(\ln{\epsilon})^2} + ...
2
votes
2answers
58 views

True of false: The sum of this infinite series. [duplicate]

I'm fairly sure it is false, but I'm not quite sure about which test I should use to prove it. $$\sum_{n=2}^\infty \ln\left(\frac{n-1}{n}\right) = -1 $$ I think using the integral test should work, ...
0
votes
1answer
28 views

Calculating a Hausdorff Dimension from Formula

I am having difficulty computing the following formula for the Hausdorff Dimension for a type of Moran set called a partial homogenous Cantor set. ...
2
votes
2answers
31 views

Common factor out from a sum of exponential functions

From the below equation, which is a sum of two exponential functions I would like to compute the common factor $n$ $$ d = \exp\left(\frac{-x}{n}\right)+\exp\left(\frac{-y}{n}\right)$$ Unfortunately, ...
1
vote
0answers
40 views

Integral of complex logarithm on a disk in the plane

Let $a$ be a complex number and $D$ the disk centered around $0$ and of radius $R$. I would like to compute the integral I=$\int_D \log(|z-a|)d^2z$. I am interested in particular in the case $R\gg ...
23
votes
5answers
1k views

Solving a logarithmic equation that has an exception to the power rule

Given the following: $$\log_3({x^2-3})^2=2$$ If I were to use the power rule, I would do: $$2\log_3({x^2-3})=2$$ $$\log_3({x^2-3})=1$$ $$3^1=x^2-3$$ $$3+3=x^2$$ $$x=\pm\sqrt6$$ Substituting ...
3
votes
0answers
47 views

Derivative of the Logarithm - Dirac

So I stumbled across P.Dirac's book Principles of Quantum Mechanics and I found something really peculiar on page 61 of the Fourth Edition. He states that usually we accept that ...
1
vote
1answer
24 views

Smart way to calculate floor(log(x))?

I thought of an algorithm that involves $\lfloor \log_{b} x \rfloor$ and am trying to determine its computational complexity. At first glance my algorithm looks polynomial, but I read that my ...
2
votes
3answers
34 views

Convergence test of $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$

Does S converge? (The answer says it converges) $S=\frac{1}{\ln 2} \sum_{k=1}^\infty \ln (1+\frac{1}{k(k+2)}) \ln k$ My attempt: Comparison test: $\ln (1+\frac{1}{k(k+2)}) \ln k \lt \ln 2 ...
2
votes
1answer
21 views

Find depth of three node tree

I am trying to write a formula to find the depth of a three node tree and having issues doing it. Each node will have an index number going from top to bottom, left to right. It will look something ...
2
votes
1answer
30 views

Is $O(n^k \log n)$ of smaller time complexity than $O(n^{k+\epsilon})$?

Is it true that asymptotically, $O(n^k \log n)$ is of smaller time complexity than $O(n^{k+\epsilon})$ for $\epsilon>0$? How might I prove this one way or the other?
3
votes
2answers
57 views

Why does it seem to be that I can raise negative numbers to the power i?

I recently encountered the ided of raising a number to the imaginary unit, and I've been trying to figure out what that means and haven't really found any useful resources. So, I came across this ...
5
votes
6answers
90 views

How is $2^{\log_4 n}= n^{\log _42}$?

I saw in a notebook the following: $2^{\log_4 n}= n^{\log _42}(=\sqrt n)$, but I never saw this before and I can't find it in any log rules, is it right? and if so how did they do it? BTW, if we take ...
1
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0answers
19 views

Developing log function always clamped at y=1 and asymptotic at x of my choosing

Math has never been my strong suit. But with my head in the books and me paying attention at every step while problem solving I did... pretty good. But my calculus was 20+ years ago. I'm close to zero ...
1
vote
5answers
56 views

How to solve this limit involving sine and log?

I've tried L'Hopital's Rule but the differentiated numerator involves cos(1/x) which does not exist when x approaches 0. $$ \lim_{x\to 0^+} \frac{x^2sin\frac{1}{x}}{\ln(1+2x)}$$
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votes
2answers
61 views

Integral of $\int\frac{1}{1+2e^x}dx$

It seems there are two ways to find the integral of this function $f(x) = \frac{1}{1+2e^x}$. In both paths I only do operations that I know are true, but for some reason one of them gives me the right ...
0
votes
1answer
36 views

solving for a variable that exist inside as well as outside of natural log or exponent

can the following equation be solved for K analytically? If not, then what other approaches I could try out? K*ln[(C2-K)/(C1-K)] = -(F/V)*t The original equation ...
2
votes
2answers
41 views

Finding the Limit by doing the Natural Log of the Numerator and Denominator

I want to show more concretely that $$\lim_{x\to\infty} \frac{e^\sqrt{x}}{x^{a}}$$ approaches $\infty$ to do this I did the natural log of the numerator and denominator and then did L'Hospital's Rule. ...
1
vote
1answer
31 views

Why is this $O(\log \log n)$?

Why is this $O(\log \log n)$? // Here c is a constant greater than 1 for (int i = 2; i <=n; i = pow(i, c)) { // some O(1) expressions } I am ...
2
votes
3answers
24 views

Prove that $|\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$

I need to show that $ \forall x,y \in \mathbb R, |\log(1 + x^2) - \log(1 + y^2)| \le |x-y|$ I tried using concavity of log function: $\log(1 + x^2) - \log(1 + y^2)=\log(\frac{1 + x^2}{1 + ...
0
votes
1answer
21 views

Logarithm and Exponent Relation [closed]

Let I have an equation $\mathcal{p} = 3^n*I$ where $I\in\{0,1,2\}$ then can I find out $I$ using $\log$ ?. Assuming $n$ is unknown. And only $p$ is shared to you.
1
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0answers
30 views

Is my proof valid for $\log(n!) = \Theta(n \log n)$?

Is my proof valid for $\log(n!) = \Theta(n \log n)$? First I prove that $\log(n!) \leq cn \log n$ for some positive $c$ for all $n \geq n_0$. Since $n! \leq n^n$, it follows that $\log(n!) \leq ...
2
votes
2answers
57 views

Is the ratio of two natural logarithms irrational or rational?

Is there any way to prove that the ratio of two natural logarithms is rational or irrational? Take the natural logarithms of $a = 25$ and $b = 6$, for example. Can you prove $\ln(a)/\ln(b)$ rational ...
0
votes
4answers
73 views

Calculate the Limit as x approaches 0

I am asked to calculate the following limit $$ \lim_{x\to0}\frac{\ln(1+\sin x)}{\sin(2x)} $$ First, I tried expressing $1+\sin x=t$, then express $x$ from that equation but my equation seemed to just ...
-2
votes
1answer
85 views

about the differentiation of $n^{1.2}\log\log\log n$

Could anyone help to resolve this question? $$\lim_{n \to \infty} {n\log n \over n^{1.2}\log\log\log n}$$ So in this question I try to use the L'Hopital's rule and do the differentiation, I could ...
1
vote
1answer
38 views

Approximate log of the sum

Suppose I want to approximate the following sum: $\log( \sum_{n=1}^\infty s_n e^{X_{n}})$, where $(X_n)$ is linear. Is there any smart way to approximate the first sum non-numerically?
1
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0answers
38 views

Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number ...
0
votes
2answers
128 views

Show that $f(x^a) = a f(x) $ and $f(x y) = f(x) + f(y)$

Let $f(x) = (x-1) \large\prod_\limits{n=1}^{\infty} \dfrac{2}{x^{2^{-n}} + 1} $ For real $x > 0$ it is easy to show that $f(x^2) = 2 f(x)$. Let $a$ be a real number. Question 1 Show that ...
-1
votes
2answers
57 views

Solve for $x$ in $\frac{x}{\ln(x)}=a$. Why does Wolfram alpha use complex numbers here?

Is there any possible way of doing this without using complex numbers? And why are complex numbers used?
1
vote
2answers
37 views

Find the value of $\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$

if $x,y,z \gt 1$ and $x^2=yz$ find the value of $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)$$ what i did is $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)=(1+4\log_{xz}y)\times (1+4\log_{xy}z)$$ ...
1
vote
2answers
34 views

Change of base proof without reciprocal

I am looking for a proof of the change of base formula without using the reciprocal. I know that: $$log_ax=\frac{log_bx}{log_ba}$$ The proof usually involves taking the reciprocal: $log_ax=y$ ...
1
vote
2answers
51 views

Are there other functions of sets $f$ such that they have this property?

$f(A \cap B) = f(A) \cup f(B)$ This function is similar to the $\log_c$ function in that application of it onto a multiplication is equivalent to the summation of its applications. $\log_c(ab) = ...
1
vote
0answers
48 views

Three almost-integers of the form $ce^{H_a+H_b}\approx 2^k\pm1$

The approximation $$H_n\approx log(2n+1)$$ http://math.stackexchange.com/a/1602945/134791 suggests that the harmonic number for composite odd numbers might be close to the sum of the harmonic ...
-5
votes
0answers
57 views

Logarithms applied to aleph 0 [duplicate]

I know that Ln(aleph 1) = aleph0 So Ln(aleph 0) = ? What is that number? Thanks for attention, really. Cheers R. Aragón M.
0
votes
3answers
22 views

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?

How to express $\log_3(2^x)$ using $\log_{10}$? And how to evaluate $4^{\log_4y}$?
0
votes
3answers
89 views

Showing that the series $\sum \log{n}/n^2$ converges.

I aim to show that the series $$\sum \frac{\log{n}}{n^2}$$ converges. I know that the inequality $log(n) < \sqrt{n}$ holds for large $n$. So this give us one way to prove the convergence of $\sum ...
0
votes
0answers
17 views

A group with unusual discrete log properties.

Does there exist a group where computing $g^x$ from $g^{a^{x}}$ is easy, computing $g^{a^{x}}$ from $x$ and $g^{a}$ is hard, and computing $x$ from $g^a$ and $g^{a^x}$ is hard. Intuitively I would ...
1
vote
1answer
48 views

Solving logarithmic equations without calculator

Hi I am stuck on this question $$ \log_x 10= 5 (\log_{10} x) +4 $$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.
0
votes
3answers
62 views

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. [closed]

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. Is there any property that can help here?
12
votes
1answer
121 views

Divergent function of ratio must be logarithm

Given. Consider two functions $F(t)$ and $r(t,x)$ such that $\lim_{t\to\infty} F(t) = \infty$ and $\lim_{t\to\infty} r(t,x)$ is finite for any $x$. ($x$ and $t$ are always positive in what follows.) ...
2
votes
4answers
60 views

Paradox: Summation of natural logarithms

Consider the expression : $$\sum_{i=1}^{\infty}\ln(i+2)-\ln(i+4)$$ If one evaluates it out, one gets $$\ln(\frac{3\times4\times5\times6\times...}{5\times6\times7\times8\times...})=\ln(12)$$ That ...
0
votes
1answer
30 views

Why does $\ln(1+\frac{3}{n^2}+o(\frac{1}{n^2}))=\frac{3}{n^2}+o(\frac{1}{n^2})$?

In order to show that a series converges, I want to show that $\sum\ln(\frac{v_{n+1}}{v_n})$ Which led me to the following first part of the equation, but I didn't achieved to solve it so I looked in ...