Questions related to real and complex logarithms.

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2
votes
1answer
53 views

$f(x)\in O(\frac{1}{x})$ implies $\log(f(x))\in O(\frac{1}{x^2})$?

Consider a function $f(x):(0, \infty) \rightarrow \mathbb{R}$. Suppose $f(x)\in O(\frac{1}{x})$ as $x\rightarrow 0$ where Big O notation is described here. Is it true that $$ \log(f(x))\in ...
2
votes
2answers
66 views

Limits problem without L'Hopital

I am prompted to solve the following limit $$\lim_{x\mapsto 0} (\cos(x)^\frac{1}{x^2})$$ I try to approach this problem by doing $$\lim_{x\mapsto 0} (-1+(1+\cos(x))^\frac{\cos(x)}{\cos(x)x^2})$$ ...
2
votes
2answers
68 views

Let $a$ be a real number satisfying $0<a<1$. Evaluate $\lim\left (\frac{a^n+a^{3n}}{1+a^2}\right)^{1/n}$

let a be a real number satisfying $0<a<1$. Evaluate $$\lim \left(\frac{a^n+a^{3n}}{1+a^2}\right)^{1/n}$$ I feel i should be doing this my squeeze so i wrote this: ...
1
vote
0answers
60 views

Axiomatic definition of exponent

Here's something I'm thinking about for a while, and would like to get feedback and some relevant references. Say I want to define by axioms an operation that will act like exponent, without using ...
1
vote
5answers
71 views

Solving $3^x \cdot 4^{2x+1}=6^{x+2}$

Find the exact value of $x$ for the equation $(3^x)(4^{2x+1})=6^{x+2}$ Give your answer in the form $\frac{\ln a}{\ln b}$ where a and b are integers. I have tried using a substitution method, i.e. ...
3
votes
0answers
48 views

The $n$th integral of $\ln(x)$ and fractional derivatives

For a related question, I need to know the $n$th integral of $\ln(x)$ and the fractional derivative of $\ln(x)$. A break down of how fractional derivatives may be found on the Wikipedia. In ...
7
votes
3answers
702 views

What's wrong with this logarithm calculation?

We know that $\displaystyle\log_a(xy)=\log_ax+\log_ay$. Consider the following: $$\displaystyle\ln(1)=\displaystyle\ln((-1)\times(-1))=\displaystyle\ln(-1)+\displaystyle\ln(-1) $$ ...
0
votes
1answer
16 views

How to write Kelvin equation in a different way

My question is about the Kelvin equation which is as follows: $$\ln(e/e_s) = \frac{2\cdot\sigma}{n\cdot k\cdot T\cdot r} $$ Keep in mind that the $e$ in $\ln(e/e_s)$ is not the constant ...
1
vote
0answers
46 views

Taylor of $\ln(f(exp(x))))$?

Let $ f(x) = \sum a_n x^n$ Such that The $a_n$ are real and $f(a),f ' (a) , f " (a) > 0 $ for any real $a > 0$. Let $ \ln(f(exp(x))) = \sum b_n x^n $. Let $c_n = a_n - b_n$. For a given $f$ ...
1
vote
0answers
69 views

Prove that $\ln(n!) = (n+\frac{1}{2})\ln(n) - n + C + o(1)$

The question: (1) Prove that $\ln(n!) = (n+\frac{1}{2})\ln(n) - n + C + o(1)$, where $C - const$, o(1) - infinitely small. (2) Using Valles' formula ...
4
votes
2answers
258 views

Are the additive group of rationals and the multiplicative group of positive rationals isomorphic? [duplicate]

This question is a little bit different from Group of positive rationals under multiplication not isomorphic to group of rationals since I was wondering if logarithmic function could solve this or ...
1
vote
2answers
37 views

Finding a solution to this logarithm equation.

How is: $ \log_{(xyz)^{1/3}} \left(\frac{yz}{x^{3}}\right)^{2}$ expressed in terms of a and b when: $ a = \log_{y} x $ $ b = \log_{z} x $ EDIT: It is z in the numerator, apologies i posted by ...
0
votes
1answer
27 views

$ \sup_{x \in A}\log(f(x))=\log(\sup_{x \in A}f(x)) $?

Consider a function $f:A\subseteq \mathbb{R} \rightarrow (0,\infty) $. Is $$ \sup_{x \in A}\log(f(x))=\log(\sup_{x \in A}f(x)) $$ true? I believe the answer is yes but I would like to have a formal ...
3
votes
3answers
49 views

Solving recurrence $T(n) = T(n-2) +2 \log(n)$ using the Substitution Method

$T(n) = T(n-2) +2 \log(n)$ if n>1 & 1 if n=1 So I start by substituting 3 times to get an idea about the pattern: $T(n)=T(n-4) + 2 \log(n-2) + 2 \log n$ $T(n)=T(n-6) + 2\log(n-4) + 2\log(n-2) + ...
0
votes
2answers
28 views

Domain of logarithmic function

We have a logarithmic function $f(x) = \log_3[(x-2)(x-3)]$. In order to determine a domain of this function we have to solve an equation $(x - 2)(x - 3) > 0$. The result is a range $(-\infty, 2) ...
2
votes
2answers
60 views

Intutively, why does $x^{\frac{1}{\ln x}} = e$?

For any $x \gt 0$, we have this odd identity: $$x^{\frac{1}{\ln x}} = e$$. You can see this by using the fact that $x = e^{\ln x}$. I'm wondering if there's a good intuitive explanation for this ...
1
vote
3answers
64 views

Convert $(\log n)^n$ to the form $n^x$ for some $x$

How do I convert ${\log n}^n$ to the form $n^x$, for some $x$? I'd like to compare the big-O runtime of $(\log n)^n$ to $n^{\log n}$ directly. Intuitively, $(\log n)^n$ grows faster since the exponent ...
1
vote
2answers
67 views

Logarithm of a positive number

I'm new to this site and I need help on this logarithm question. I don't know how to approach this question to simplify it. $$\log_2(x^2-4)−3\log_2\frac{(x+2)}{(x-2)}>2$$ Apparently the answer is ...
1
vote
0answers
35 views

It is possible a Skewes number between twin primes? Can you discard such extreme question?

Let $p_n$ the nth prime number. A prime number $p=p_{k}$ is called twin prime if $2+p_k$ is also a prime number. Since $1+p_k$ is even then we have obviously that $2+p=p_{k+1}$. My purpose with this ...
0
votes
1answer
31 views

calculating logarithmic equation

I have an equation , which is like this $64n\log n < 8n^2$ . (the base of logarithm is 2) I know how to solve the logarithmic equations . I am a programmer , so I wrote a simple program and ...
2
votes
1answer
58 views

Proof of concavity of log function

Does anybody have a proof of the concavity of the $\log{x}$ that does not use calculus?
0
votes
2answers
33 views

Simplification of Log Factorial Expression

I would like to find a simplification of the expression \begin{equation} \log{\frac{(x+y+z)!}{x! y! z!}} \end{equation} that is linear with respect to $x, y$, and $z$. Does such an expression exist? ...
3
votes
5answers
41 views

Limit of function of hyperbolic

How can I - without using derivatives - find the limit of the function $f(x)=\frac{1}{\cosh(x)}+\log \left(\frac{\cosh(x)}{1+\cosh(x)} \right)$ as $x \to \infty$ and as $x \to -\infty$? We know ...
5
votes
3answers
873 views

Is it possible to find inflection points by setting the first derivative to 0?

I have the following $$y = \frac{x^2}{2}-\ln x$$ $$y'= x - \frac1x$$ I learned that inflection points were found by setting the $2^{nd}$ derivative equal to $0$, however, if I do that in this case ...
0
votes
0answers
38 views

True or False: $f(z)=Ln(z)$ is periodic

From Wolfram MathWorld: A function $f(x)$ is said to be periodic with period $p$ if $f(x)=f(x+np)$ for $n=1,2,3...$ The easy bait here is to realize that $Ln(z)$ is asking for the principal ...
0
votes
2answers
31 views

Find $x = \log_2(4^{3\over 4}\cdot \sqrt{2^5})^{1 \over 2}$

I am looking for $x$. $$x = \log_2 \left[(4^{3\over 4}\cdot \sqrt{2^5})^{1 \over 2}\right]$$ I am not sure how to do this. I am trying to solve this by changing the form like this: $$x = \log_a b ...
0
votes
0answers
61 views

Show that $(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ [duplicate]

$(\ln 6)^{(\ln 5)^{(\ln 4)^{(\ln 3)^{(\ln 2)}}}}<\pi$ How can we show this equivalent without using a calculator?
-1
votes
1answer
40 views

Inverse/integral/limit of logarithmic function

I have this problem: $f:(0,\infty)\rightarrow \mathbb R$, $$f(x)=x+1+\ln x $$ $g$ is the reverse of $f$. The problems are: $g'(2) =$? and $$\int_{2}^{e+2}g(x)dx$$ and ...
0
votes
3answers
39 views

Inverse of logarithmic function

I have this problem: $f:(0,\infty)\rightarrow \mathbb R$, $$f(x)=x+1+\ln x $$ $g$ is the reverse of $f$. I tried to solve the problem like this $y=f(x) ; x+\ln x=y-1$ and I got stuck The problem ...
-2
votes
5answers
47 views

Logarithm computation (a fraction) [closed]

I am trying to compute the following fraction but get failed as it leads me to log over log. $$\frac{\log_{10}0.04-2\log_{10}0.3}{1-\log_{10}15}=2$$ Please help.
5
votes
0answers
84 views

An integral to prove that $\log(2n+1) \ge H_n$

Dalzell integral The equation $$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ proves that $\frac{22}{7}-\pi>0$ because the integrand is positive. Some Dalzell-type integrals for ...
1
vote
1answer
44 views

Finding the domain $y=\log_{2x}(7)$

The question is to find the domain of $y=\log_{2x}(7)$ What I have attempted: $$y=\log_{2x}(7)$$ $$ y = \frac{\log(7)}{\log(2x)}$$ $$ y = \frac{\log(7)}{\log(2)+\log(x)}$$ The ...
1
vote
1answer
75 views

Does the logarithm function grow slower than any polynomial?

Does $f(x)=ax^b$ grow faster than $g(x)=\ln x$ for all $a, b > 0$? Can I say that $f(x) > g(x)$ as $x$ approaches infinity? I thought the answer is yes, but this graph appears to be telling a ...
1
vote
6answers
66 views

How do I calculate the limit of this log function?

I've been stuck with calculating the limit of the following problem for a while now. Can you help? $$ \lim_{x\to \infty} \frac{\sqrt{\log(x) + 1}}{\log(\log(x))} = $$
0
votes
3answers
36 views

Evaluate the limit $\lim_{x \to 0}\frac{-zx \ln (zx)}{-x \ln x}$

Title says it all. How can I evaluate the limit provided as: \begin{equation} \lim_{x \to 0^+}\frac{-zx \ln (zx)}{-x \ln x} \end{equation} I do know that it is equal to $z$. What I do not know is ...
1
vote
0answers
29 views

Calculation of logarithms [duplicate]

I understand the properties of logarithms. However I am curious if a calculator uses complex power series, slide rule program, Taylor series, or a form of an algorithm to solve for log"base"n(x).
0
votes
1answer
14 views

Find the value of $k$ in the exponent

I am trying to calculate the $k$ value in this equation: $\dfrac1{n^c} \le \left(1 - \dfrac2{n(n-1)} \right)^k$ by using the logarithm, I am getting for $k$: $\log_{1- 2/n(n-1)} n^{-c} \le k$ is ...
1
vote
1answer
19 views

Simplifying trivial expression

How to simplify an expression: $e^{\frac{-\ln^2 y}{2}}$? I mean because as if $e^{\ln y}=y$ then my guess is that this could be simplified in a similar manner thus how to do that?
0
votes
2answers
37 views

How can I solve this nature log equation?

$ln(x+2)=e^{(x-4)}$ Is there any way to solve this equation without graphing or using GDC ? Thank you
4
votes
2answers
94 views

Which meromorphic functions are logarithmic derivatives of other meromorphic functions?

Let $f$ be a meromorphic function defined on the whole complex plane. Is there a characterization in terms of easier-to-test properties of $f$ whether or not $f=g'/g$ for some entire $g$? The ...
1
vote
1answer
57 views

When can I assume $\operatorname{Log}$ means the natural log?

I have a problem that says: Express $e^{\operatorname{Log}(3+4i)}$ in the form $x+iy$. However, in class we usually only deal with natural log (this problem is from a textbook). How do I know when I ...
0
votes
1answer
11 views

Masters Theorem constant K and Polynomial Difference

In my lecture notes I noticed for case 2 of masters theorem there is constant "k". Where does that constant come from? Also if f(n) and (n^log b a) is not polynomial different from each other, why ...
0
votes
0answers
19 views

Question regarding plotting 120,000 points uniformly using log scaling

I have a bit of a strange problem I can't seem to wrap my head around. I am developing a game for the phone as a home project, I have a data-set of 120,000 different 3D points which represent the ...
0
votes
2answers
33 views

Is there a way to simplify $a\log(b) +\log(c)$?

Very simply: $$a\log(b) +\log(c)$$ The coefficient on the first term is throwing me off.
2
votes
2answers
37 views

Can you multiply a natural log term by a constant to take power inside log term?

For Calculus 2 homework I must prove that $\displaystyle\int \frac{du}{u^2-a^2} = \frac{1}{2a}\ln\left|\frac{u-a}{u+a}\right|+C$ The instructor wants us to use trigonometric substitution to solve. ...
2
votes
5answers
64 views

$\ln^2(x)\overset{?}=2\ln(x)$

Is it the same as $\ln(x)^2$? And if so, is it equal to $2\ln(x)$? Thanks in advance.
2
votes
1answer
114 views

Series and integrals for inequalities and approximations to $\log(n)$

The following series and integrals relate log(2) to its third and fourth convergents, $\frac{2}{3}$ and $\frac{7}{10}$. $$\begin{align} \log\left(2\right)-\frac{2}{3} &= \sum_{k=1}^\infty ...
1
vote
1answer
30 views

Astronomical magnitude formula convertion

It is a question asked out of pure frustration, but I really want to know the steps leading from the expression $$m_1-m_2=2.5\log\bigg(\frac{F_2}{F_1}\bigg)$$ to this one ...
0
votes
1answer
32 views

Determine logarithm versus known values of $ a,b$ [closed]

We know $\log_{30}(3)=a, \log_{30}(5)=b.$ How to determine $\log_{30}(16)$?
25
votes
8answers
3k views

Easy way to compute logarithms without a calculator?

I would need to be able to compute logarithms without using a calculator, just on paper. The result should be a fraction so it is the most accurate. For example I have seen this in math class ...