Questions related to real and complex logarithms.

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9
votes
5answers
629 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.
6
votes
3answers
258 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 ...
7
votes
4answers
820 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 ...
17
votes
5answers
1k 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 ...
22
votes
10answers
1k 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, ...
8
votes
5answers
1k 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 ...
0
votes
1answer
573 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 ...
65
votes
16answers
9k 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?
23
votes
3answers
6k views

What algorithm is used by computers to calculate logarithms?

I would like to know how are logarithms calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that ...
32
votes
4answers
1k views

Integral $\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx$

Is there a closed form for the integral $$\int_0^1\frac{\ln\left(x+\sqrt2\right)}{\sqrt{2-x}\,\sqrt{1-x}\,\sqrt{\vphantom{1}x}}dx.$$ I do not have a strong reason to be sure it exists, but I would be ...
47
votes
4answers
2k views

Prove $\left(\dfrac{2}{5}\right)^{\frac{2}{5}}<\ln{2}$

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 ...
46
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?
13
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 ...
8
votes
1answer
222 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
77 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) ...
14
votes
11answers
931 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 ...
15
votes
3answers
2k 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
1answer
768 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} ...
6
votes
2answers
899 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. ...
6
votes
6answers
1k 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 ...
6
votes
2answers
1k 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
402 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
233 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 ...
3
votes
4answers
298 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 ...
2
votes
0answers
60 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
3answers
126 views

Write the expressoin in terms of $\log x$ and $\log y \log(\frac{x^3}{10y})$

What is the answer for this? Write the expression in terms of $\log x$ and $\log y$ $$\log\left(\dfrac{x^3}{10y}\right)$$ This is what I got out of the equation so far. the alternate form assuming ...
1
vote
4answers
309 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 ...
27
votes
1answer
657 views

Prove $\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$

How can I prove the following identity? $$\int_0^1\frac{x^2-2\,x+2\ln(1+x)}{x^3\,\sqrt{1-x^2}}dx=\frac{\pi^2}8-\frac12$$
20
votes
5answers
2k views

Fun logarithm question

How would you go about solving $x^{2^{x}} = 2^{x}$? There should be a solution $1<x<2$, but I haven't found a way to derive the answer using the usual log laws, maybe there is an elegant way ...
9
votes
3answers
352 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 ...
1
vote
4answers
796 views

Why aren't logarithms defined for negative $x$?

Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
14
votes
3answers
518 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
433 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 ...
3
votes
5answers
174 views

How to prove that $\ln(x)<x$ for $x\to \infty$?

During my calculus homework I need to prove some limits without using L'Hôpital's rule. I have difficulties to show rigorously that $\ln(x)<x$ for big enough x. For example, I need to find the ...
5
votes
1answer
797 views

How to figure of the Laplace transform for $\log x$?

I was looking at a table of common Laplace transforms of functions when I came across the transform for $\log x$. Apparently, the transform is as follows: $$\mathcal{L} \left\{ \log ...
4
votes
1answer
488 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
1
vote
6answers
200 views

How to find $\lim\limits_{n\rightarrow \infty}\frac{(\log n)^p}{n}$

How to solve $$\lim_{n\rightarrow \infty}\frac{(\log n)^p}{n}$$
1
vote
1answer
460 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 ...
19
votes
2answers
549 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 ...
8
votes
6answers
414 views

Has anyone talked themselves into understanding Euler's identity a bit?

What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
4
votes
3answers
809 views

Find a branch of $f(z)= \log(z^3-2)$ that is analytic at $z=0$.

Find a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$. Can anyone help me on this question? I have no idea how to find a branch. The definition of branch given in lecture is $F$ is a branch ...
4
votes
1answer
295 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
761 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.
2
votes
1answer
156 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.
1
vote
1answer
19 views

how to graph logarithmic and exponential equations

I am not looking for the answer to the question, it would be helpful but an explanation would also be very helpful. How would you graph the following equations? ln is log base e, while log is log ...
1
vote
0answers
33 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
1answer
138 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 ...
0
votes
0answers
49 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
39 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 ...
29
votes
1answer
838 views

Closed form for $\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx$

Please help me to find a closed form for the following integral: $$\int_0^1\log\left(\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\right)\,{\mathrm d}x.$$ I was told it could be calculated in a ...