Questions related to real and complex logarithms.

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2
votes
2answers
36 views

Solve for $x\quad \log_2(2^n) = \log_2(1+x)$

I am out of practice with logs, but this is derived from the channel capacity theorem. $$B\log_2\left(1 + \frac SN\right)$$ Solve for $x $ $$\log_2(2^n) = \log_2(1+x)$$ I need this equation ...
0
votes
3answers
126 views

Exponential (to the power of a logarithm) [closed]

How do I solve the following equation: $(3x)^{ln3}=(4x)^{ln4}$ Thanks in advance!
-2
votes
2answers
59 views

find the value of $k$ in the term $2^{-k} = 1/n$

What is the value of $k$ if I have the following equation: $2^{-k} = \frac1n$? $$2^{-k} = \frac 1 n \implies n = 2^k \implies \log_{2} n = k$$ Is my solution correct?
1
vote
1answer
57 views

Derivative of matrix logarithm with respect to matrix

I saw in this post that $\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$ Is this true to say: $\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$ where U is ...
0
votes
2answers
28 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
1
vote
2answers
48 views

taking the natural log of e^(2x) = (4/3)

I have been unable to answer the following question. I must solve for x: $$e^{2x} = (4/3)$$ I have been made aware that I must take the natural log of both sides, giving: $$ln(e^{2x}) = ln(4/3)$$ ...
1
vote
2answers
90 views

Use $\log(x)$ to calculate $\log(x+1)$

Given that I know the value of $\log(x)$, I would like to calculate the value of $\log(x+1)$ on a computer. I know that I could use the Taylor expansion of $\log(1+x)$, but that uses $x$ rather than ...
-1
votes
1answer
26 views

Logspace() in Matlab [closed]

In Matlab , Logspace() Generate logarithmically spaced vector . But what do we mean by them ? Earlier , I used to think that they are just equal spaced and their 10th power is returned . Like here : ...
0
votes
3answers
56 views

Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
1
vote
1answer
14 views

Log transformations of function domain and inequalities

If I know that for some function $f$, the following is true for $x, y \geq 0$: $f(\log (x^a y^b)) \leq f(\log x)^a f(\log y)^b$ Can I make the claim that $f(x^a y^b) \leq f(x)^a f(y)^b$ If I ...
3
votes
7answers
82 views

Evaluating $\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$

I am trying to find$$\lim_{n\to\infty}{n\left(\ln(n+2)-\ln n\right)}$$ But I can't figure out any good way to solve this. Is there a special theorem or method to solve such limits?
0
votes
0answers
21 views

Applying Cauchy-Riemann to $f(z)$

$$\ln|z|+i\text{Arg}(z)$$ the problem states that I have to apply Cauchy Riemann to the problem and determine a conclusion. Below is how far I got, but I'm not sure how to take the derivative of ...
0
votes
0answers
25 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
50 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) = ...
0
votes
0answers
42 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
vote
0answers
48 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
votes
2answers
55 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
vote
2answers
46 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
43 views

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

$$\sum_{k=1}^{n}\left \lfloor \log _{m}k \right \rfloor$$ $$\sum_{k=1}^{n}\left \lceil \log_{m}k\right \rceil$$ I found myself stuck trying to solve these two summations but i can't make any ...
0
votes
0answers
27 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
65 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
29 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
43 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
50 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
57 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
34 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
38 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
26 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
37 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
61 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
92 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
vote
0answers
20 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
63 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. $$ \lim_{x\to 0^+} \frac{x^2sin\frac{1}{x}}{\ln(1+2x)}$$
-1
votes
2answers
66 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
50 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
66 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
32 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 ...
3
votes
3answers
30 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
22 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
vote
0answers
34 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
61 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
90 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
39 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
vote
0answers
40 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
133 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
60 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
40 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 $$E=\log_{xz}(xy^4z) \times \log_{xy}(xyz^4)=(1+4\log_{xz}y)\times (1+4\log_{xy}z)$$ ...