Questions related to real and complex logarithms.
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?
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 ...
33
votes
3answers
609 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 ...
20
votes
0answers
392 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 $x \ge 1331$, ...
19
votes
18answers
2k views
How to understand why $x^0 = 1$, where $x$ is any real number?
Alright, so the idea of an exponent, $x$, is that you are multiplying its base by itself $x$ number of times. With base $5$ and $x=3$, we have that $5^3$ = $5 \cdot 5 \cdot 5$
I understand that the ...
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, ...
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 ...
18
votes
3answers
2k 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 ...
17
votes
2answers
266 views
Why is $\log_{-2}{4}$ complex?
With the logarithm being the inverse of the exponential function, it follows that $\log_{-2}{4}$ should equal $2$, since $(-2)^2=4$. The change of base law, however, implies that ...
17
votes
1answer
238 views
Approximation of $\log(x)$ as a linear combination of $\log(2)$ and $\log(3)$
I wonder if it's possible to approximate $\log(n)$, n integer, by using a linear combination of $\log(2)$ and $\log(3)$.
More formally, given integer $n$ and and real $\epsilon>0$, is it always ...
17
votes
3answers
504 views
Do you know of any Calculus text defining the exponential and logarithm functions in an alternative way?
The story in nearly every introductory Calculus book is well known by everybody: you don't have the "right" to raise a number to an irrational power, so forget exponents for now and let's take a look ...
17
votes
0answers
82 views
Closed form for $\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx$
Consider the following integral:
$$\mathcal{I}(\mu,\nu)=\int_0^\infty\log\frac{J_\mu(x)^2+Y_\mu(x)^2}{J_\nu(x)^2+Y_\nu(x)^2}\mathrm dx,$$
where $J_\mu(x)$ is the Bessel function of the first kind:
...
16
votes
10answers
984 views
Find the integer closest to $\ln(2013)$
I encounter such a problem, in a Maths contest, to find out the closest integer to $\ln(2013)$, without using a calculator. I really get stuck.
I tried to turn $\ln(2013)$ into ...
16
votes
6answers
427 views
$\log_9 71$ or $\log_8 61$
I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
16
votes
2answers
253 views
$2^x - a$ touches $\log_2(x)$
I was playing around with the functions $2^x$ and $\log_2(x)$. As they are the inversions of each other, I thought there was a simple number $a$ for which $2^x - a$ touches $\log_2(x)$.
Using ...
16
votes
3answers
531 views
Why isn't a harp in a logarithmic shape?
I was watching a harp, yesterday, and thought about the mathematics involved. I know that music is closely related to logarithms, because having a string or pipe twice as long produces the same note.
...
16
votes
1answer
1k views
Where did the word “logarithm” come from?
Where did the word logarithm come from? Any relation to the word algorithm?
15
votes
3answers
608 views
When log is written without a base, is the equation normally referring to log base 10 or natural log?
For example, this question presents the equation
$$\omega(n) < \frac{\log n}{\log \log n} + 1.4573 \frac{\log n}{(\log \log n)^{2}},$$
but I'm not entirely sure if this is referring to log base ...
15
votes
7answers
2k views
How did the notation “ln” for “log base e” become so pervasive?
Wikipedia sez:
The natural logarithm of x is often written "ln(x)", instead of log_e(x) especially in disciplines where it isn't written "log(x)". However, some mathematicians disapprove of this ...
15
votes
1answer
201 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
14
votes
5answers
488 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 ...
13
votes
8answers
774 views
Why does the logarithm require a special notation?
Since the logarithm is the reversed exponentiation, why does it need a distinct notation for it? Why can't we just ask:
$$2^x=8$$
Instead of:
$$\log_2 8=x$$
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 ...
13
votes
3answers
321 views
Find $x$ from $3^x\cdot x^3 = 1$
I saw a question on internet, tried to solve but I can't:
\begin{equation}
3^x\cdot x^3 = 1
\end{equation}
I get $\ln$ function and made some equalization and I reached that:
\begin{equation}
...
13
votes
3answers
367 views
Are the logarithms in number theory natural?
I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them?
Has it maybe ...
13
votes
1answer
124 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:
$$\int_0^1\log\log\left(\frac{1}{x}+\sqrt{\frac{1}{x^2}-1}\right)\mathrm dx.$$
I was told it could be calculated in a closed form.
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 ...
12
votes
3answers
989 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 ...
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 ...
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..
$$I=\frac{1}{2}\int_{0}^{\frac{\pi }{2}}\frac{\ln(\sin y)\ln(\cos y)}{\sin y\cos y}dy$$
11
votes
1answer
66 views
Closed form for $\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm dx$
I encountered this integral in my calculations:
$$\int_0^\infty\frac{\log\left(1+\frac{\pi^2}{4\,x}\right)}{e^{\sqrt{x}}-1}\mathrm ...
10
votes
3answers
495 views
What is the x in $\log_b x$ called?
In ba = x, b is the base, a is the exponent and x is the result of the operation. But in its logarithm counterpart, logb(x) = a, b is still the base, and a is now the result. What is x called here? ...
10
votes
5answers
2k views
Calculate logarithms by hand
I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits.
By pen and paper that is. I'm doing this old school.
What first came to mind was to use $\log(ab) = ...
9
votes
4answers
357 views
How to know if $\log_78 > \log_89$ without using a calculator?
I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values ...
9
votes
1answer
307 views
Is the difference of the natural logarithms of two integers always irrational or 0?
If I have two integers $a,b > 1$. Is
$\ln(a) - \ln(b)$
always either irrational or $0$. I know both $\ln(a)$ and $\ln(b)$ are irrational.
9
votes
7answers
391 views
Proving limit with $\log(n!)$
I am trying to calculate the following limits, but I don't know how:
$$\lim_{n\to\infty}\frac{3\cdot \sqrt{n}}{\log(n!)}$$
And the second one is
$$\lim_{n\to\infty}\frac{\log(n!)}{\log(n)^{\log(n)}}$$
...
9
votes
1answer
358 views
Is there a binary spigot algorithm for log(23) or log(89)?
The Bailey-Borwein-Plouffe formula yields a binary spigot algorithm for π, and related formulas give the bits of log(2) and those of the logarithms of some other integers. I got stuck (over a year ...
8
votes
9answers
431 views
Show that, for $t>0$, $\log t$ is not a polynomial.
How can I show that? I've tried to reverse the logarithm to it's exponential form in a trial to show that but I got no success. Can you help me?
8
votes
5answers
173 views
Solving an equation with a logarithm in the exponent
I try to solve the following equation:
$$ (N+1)^{\log_N{125}} = 216 $$
I know the answer is 5 here but how could I rewrite the equations so I can solve it?
I tried to take the log of both sides but ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
8
votes
4answers
589 views
Inverse of the natural log function $y =\ln x$
Of course, it is a well known fact that the inverse of $y=\ln x$ (natural logarithm of x) is $e^x$.
Assuming we haven't heard of the exponential function at all, how do we prove that the inverse of ...
8
votes
1answer
63 views
Closed form for n-th anti-derivative of $\log x$
Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
8
votes
1answer
64 views
Closed form for $\sum_{n=1}^\infty\frac{\cos(\pi \log n)}{n^2}$
Is there a closed form for the following sum? $$\sum_{n=1}^\infty\frac{\cos(\pi\log n)}{n^2}$$
8
votes
1answer
229 views
Why is $10\frac{\exp(\pi)-\log 3}{\log 2}$ almost an integer?
I read that
$$10\frac{\exp(\pi)-\log 3}{\log 2} =318.000000033252\dots \approx 318$$
Is this simply a coincidence or can this somehow be explained?
8
votes
2answers
200 views
The Fermat prime 257 and binomial sum $\sum_{n=0}^\infty \frac{(-1)^n}{\binom {8n}{4n}}$?
We have,
$\begin{aligned}
\sum_{n=0}^\infty \frac{(-1)^n}{\binom n{n/2}} &= \frac{4}{27}(9-\pi\sqrt{3}\,)\\[2.5mm]
\sum_{n=0}^\infty \frac{(-1)^n}{\binom {2n}n} &= \frac{4}{5} - ...
8
votes
1answer
217 views
Information on the sum $\sum_{n=1}^\infty \frac{\log n}{n!}$
In my personal study of interesting sums, I came up with the following sum that I could not evaluate:
$$\sum_{n=1}^\infty \frac{\log n}{n!} = 0.60378\dots$$
I would be very interested to see what ...
8
votes
0answers
217 views
Understanding Ramanujan's approach in his proof of Bertrand's Postulate
I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$
What would be wrong with this approach for ...
7
votes
4answers
390 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 ...
7
votes
4answers
583 views
Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.
Prove that if $a^x=b^y=(ab)^{xy}$, then $x+y=1$.
How do I use logarithms to approach this problem? Any help would be appreciated, thanks.
