Questions related to real and complex logarithms.
7
votes
5answers
462 views
Solve the equation $2^x=1-x$
Solve the equation:
$$2^x=1-x$$
I know this is extremely easy and I know the solution using graphical approach. Basically, I can see the solution, but I can't work it out algebraically.
19
votes
9answers
829 views
Intuitive use of logarithms
I am trying to gain a more intuitive feeling for the use of logarithms.
So my question: what do you use them for? Why were they invented? What are typical situations where one should think: "hey, ...
6
votes
3answers
122 views
Apparently cannot be solved using logarithms
This equation clearly cannot be solved using logarithms.
$$3 + x = 2 (1.01^x)$$
Now it can be solved using a graphing calculator or a computer and the answer is $x = -1.0202$ and $x=568.2993$.
But ...
6
votes
5answers
655 views
Understanding imaginary exponents
Greetings!
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to ...
14
votes
5answers
497 views
Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$
Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem.
Given hint: consider $f(z) = \ln ( 1 +z)$.
EDIT:: I know how to evaluate it, but I am ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
0
votes
2answers
58 views
Explanation needed on this rather basic recurrence solution
We are studying about recurrences in our analysis of algorithms class. As an example of the substitution method (with induction) we are given the following:
$$T(n) = \lbrace 2T\left(\frac{n}{2}\right) ...
59
votes
16answers
6k views
How do you explain the concept of logarithm to a five year old?
Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
43
votes
7answers
1k views
What's so “natural” about the base of natural logarithms?
There are so many available bases. Why is the strange number $e$ preferred over all else?
Of course one could integrate $\frac{1}x$ and see this. But is there more to the story?
5
votes
4answers
321 views
Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$
How would I go about evaluating this integral?
$$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$
What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
12
votes
2answers
1k views
Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)
I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
12
votes
3answers
991 views
How does e, or the exponential function, relate to rotation?
$e^{i \pi} = -1$. I get why this works from a sum-of-series perspective and from an integration perspective, as in I can evaluate the integrals and find this result. However, I don't understand it ...
6
votes
2answers
606 views
Is the natural log of n rational?
It's famously unknown whether the natural log of 2 is rational or not.
How about the natural log of other numbers. Is it known/unknown whether these are rational?
Obviously ln(1) is 0, and ln(2^n) ...
5
votes
2answers
338 views
Logarithm of a Markov Matrix
Start with a Markov matrix $\mathbf{M}$, whose elements are all between $0 \le \mathbf{M}_{ij} \le 1$ and each row sums to one. There is a natural connection with this matrix and the rate matrix ...
5
votes
3answers
216 views
Summing up the series $a_{3k}$ where $\log(1-x+x^2) = \sum a_k x^k$
If $\ln(1-x+x^2) = a_1x+a_2x^2 + \cdots \text{ then } a_3+a_6+a_9+a_{12} + \cdots = $ ?
My approach is to write $1-x+x^2 = \frac{1+x^3}{1+x}$ then expanding the respective logarithms,I got a series ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
4
votes
6answers
521 views
An alternative way to calculate $\log(x)$
How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$?
I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
0
votes
1answer
417 views
About irrational logarithms
Could someone provide, please, a proof of the theorem below?
"Being x and b integers greater than 1, which can not be represented as powers of the same basis(positive integer) and integer exponent, ...
33
votes
3answers
623 views
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$
Inadvertently, I find this interesting inequality,But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find ...
12
votes
11answers
666 views
Alternative notation for exponents, logs and roots?
If we have
$$ x^y = z $$
then we know that
$$ \sqrt[y]{z} = x $$
and
$$ \log_x{z} = y .$$
As a visually-oriented person I have often been dismayed that the symbols for these three operators ...
13
votes
3answers
399 views
challenging alternating infinite series involving $\ln$
I ran across an infinite series that is allegedly from a Chinese math contest.
Evaluate:
$\displaystyle\sum_{n=2}^{\infty}(-1)^{n}\ln\left(1-\frac{1}{n(n-1)}\right).$
I thought perhaps this ...
7
votes
4answers
391 views
Please help me to show, that $(\ln x)'=\frac1 x$
In school, we recently started with derivations. I looked into a list of simple derivations and tried to prove them, in order to practice. Now, I tried to find the derivative of $\ln x$, but I got ...
5
votes
3answers
220 views
Can $\displaystyle\lim_{h\to 0}\frac{b^h - 1}{h}$ be solved without circular reasoning?
In many places I have read that $$\lim_{h\to 0}\frac{b^h - 1}{h}$$ is by definition $\ln(b)$. Does that mean that this is unsolvable without using that fact or a related/derived one?
I can of course ...
3
votes
2answers
503 views
Motivation for Napier's Logarithms
In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this section. ...
3
votes
4answers
264 views
How $a^{\log_b x} = x^{\log_b a}$?
This actually triggered me in my mind from here. After some playing around I notice that the relation $a^{\log_b x} = x^{\log_b a}$ is true for any valid value of $a,b$ and $x$. I am very inquisitive ...
1
vote
1answer
392 views
Why can't you integrate all power functions without a log function?
You need a logarithm function to solve all power functions. That's a fact.
Power functions look like this:
$f\colon x \mapsto a x^r \qquad a,r \in \mathbb{R}$
But why would you need a logarithm ...
18
votes
2answers
490 views
$\ln(x^2)$ vs $2\ln x$
These two are supposed to be equivalent because of the properties of logarithms, but the domains of $\ln(x^2)$ and $2\ln x$ seem different to me. For example, if I substitute $x=-1$ into the first, I ...
4
votes
1answer
216 views
The definition of the logarithm.
One usually gets several definitions of the logarithm along his studies.
You might be first introduced to the exponential and then told that the logarithm is its inverse.
You might be given
$$\log ...
2
votes
3answers
208 views
How can I solve for $n$ in the equation $n \log n = C$?
Believe it or not, this isn't homework. It's been many years since grade school, and I'm trying to brush up on these things. But my intuition isn't helping me here.
1
vote
0answers
28 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
1
vote
4answers
271 views
Finding all functions $f$ satisfying $f'(t)=f(t)+\int_a^bf(t)dt$
I am trying to find all functions f satisfying $f'(t)=f(t)+\int_a^bf(t)dt$.
This is a problem from Spivak's Calculus and it is the chapter about Logarithms and Exponential functions. I gave up ...
1
vote
1answer
106 views
Can you use a logarithm coefficient in a linear equation?
I have an equation that looks like $x+(\ln3)y+z=0$ where there's a natural logarithm as a coefficient. Is it possible to have this in a linear equation? I know that you cannot have a root or a product ...
1
vote
1answer
134 views
how to find the value of $\log_3 7$
Can I ask how to compute $\log_3 7$, using the changing the base of logarithm.
0
votes
0answers
41 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
0
votes
1answer
37 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
11
votes
4answers
339 views
How to calculate $I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$?
How do I integrate this guy? I've been stuck on this for hours..
$$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
5
votes
2answers
91 views
How to formally show that $f(z)$ is analytic at $z=0$?
Let $z$ be a complex number. Let $$f(z)=\dfrac{1}{\frac{1}{z}+\ln(\frac{1}{z})}.$$ How to formally show that $f(z)$ is analytic at $z=0$?
I know that for small $z$ we have ...
8
votes
1answer
229 views
Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?
I read that
$$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$
Is this simply a coincidence or can this somehow be explained?
6
votes
1answer
332 views
How many digits does $2^{1000}$ contain?
I tried this way, I only need to know if this is correct or if there are better ways to solve this:
$2^{1000}$ does not have a factor of $5$ obviously therefore we can assume
$$ 10^{m} < 2^{1000} ...
2
votes
5answers
484 views
1
vote
5answers
146 views
Limit of a recursively defined bivariate function.
Let m and n be positive integers.
Let $f(m,0)=m$
Let $f(m,n)= e \ln(f(m,n-1))$
$$\lim_{m\to\infty} \ln(m)\Big(f(m,\lfloor\ln m\rfloor)) - e\Big) = 163^{1/3}+C$$
Where $C$ is a constant.
It seems ...
6
votes
4answers
209 views
Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$
I have this exponential equation that I don't know how to solve:
$3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$
I tried to factor out a term, but it does not help. ...
4
votes
1answer
46 views
Looking for help in understanding Jitsuro Nagura's analysis of the upper bound for $\psi(x)$
I'm working on understanding Nagura's analysis of the upper bound for $\psi(x)$ which is done in Lemma 2. I am unclear on one step of his reasoning.
With Lemma 1, he establishes for $x \ge 2000$:
...
3
votes
1answer
54 views
Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?
In Ramanujan's proof of Bertrand's Postulate, Ramanujan states:
$\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$
where:
$\vartheta(x) = \sum_{p \le x} ...
3
votes
4answers
253 views
Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$
Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$
How to prove without a computer?
2
votes
3answers
168 views
How can I solve $8n^2 = 64n\,\log_2(n)$
I currently try to analyze the runtime behaviour of several algorithms.
However, I want to know for which integral values $n$ the first algorithm is better ($f(n)$ is smaller) and for which the second ...
2
votes
2answers
2k views
The difference between log and ln
$$\dfrac{1}{2}\ln(x+7)-(2 \ln x+3 \ln y)$$
Our professor let's us solve this but i do not understand how $\ln$ works. He says it has same properties with $\log$ but i still don't get it. What's the ...
2
votes
1answer
280 views
Why does an equiangular spiral become logarithmic (intuitively)?
One of the most famous 2D-curves are logarithmic spirals (or Spira mirabilis). They can be constructed by using a machinery that ensures a constant angle between the tangent and the radial lines all ...
20
votes
0answers
394 views
Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$
Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
