Questions related to real and complex logarithms.

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8
votes
3answers
221 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
8
votes
3answers
477 views

Broken Calculator: only certain unary functions work.

I have run into a challenge on Codecademy.com that has me absolutely bewildered. I'm sure I'm just overlooking an obvious solution, but I've been scouring tables of trigonometric and logarithmic ...
8
votes
2answers
244 views

Conjecture $\int_0^1\frac{\ln^2\left(1+x+x^2\right)}x dx\stackrel?=\frac{2\pi}{9\sqrt3}\psi^{(1)}(\tfrac13)-\frac{4\pi^3}{27\sqrt3}-\frac23\zeta(3)$

I'm interested in the following definite integral: $$I=\int_0^1\frac{\ln^2\!\left(1+x+x^2\right)}x\,dx.\tag1$$ The corresponding antiderivative can be evaluated with Mathematica, but even after ...
8
votes
2answers
130 views

Closed form for $\int z^n\ln{(z)}\ln{(1-z)}\,\mathrm{d}z$?

Problem. Find an anti-derivative for the following indefinite integral, where $n$ is a non-negative integer: $$\int z^n\ln{\left(z\right)}\ln{\left(1-z\right)}\,\mathrm{d}z=~???$$ My attempt: ...
8
votes
1answer
266 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
5k 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
2answers
258 views

Prove $ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2 $

Prove that $$ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$ where last ...
8
votes
4answers
173 views

What is the inverse of $2^x$? [duplicate]

Note: This may not be correct mathematical term, so in case of confusion, I mean what division is to multiplication. If not, just poke me in the comments. I was given this the other day: $2^x=8$ ...
8
votes
2answers
273 views

Integral $ \int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$

Please help evaluating this integral $$ \large\int_{0}^1 \sqrt{\frac{\ln{x}}{x^2-1}} dx$$ Mathematica could not evaluate it in a closed form. Numerically it is about $$I\approx1.060837861412045137097....
8
votes
2answers
1k views

Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? I'...
8
votes
1answer
837 views

Inverse function of $\operatorname{li}(x)$ over $x>\mu$?

How can I get the inverse function of $\operatorname{li}(x)$ over $x>\mu$? Where $$\operatorname{li}(x)=\int_{0}^{x}\frac{ds}{\ln(s)}$$ is the so-called logarithmic integral, and $\operatorname{li}...
8
votes
1answer
283 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
1answer
81 views

Evaluate $\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$

Is there a closed form of $$\lim_{R\to\infty}\left(\int_0^R\left|\frac{\sin x}{x}\right|dx-\frac{2}{\pi}\log R\right)$$ I am pretty interested whether we can find out a closed form of this limit. We ...
8
votes
1answer
245 views

Feynman's Algorithm for computing a logarithm of a number in [1,2]

I came upon the following quote concerning Feynman (the entire essay this is from can be found here): Consider the problem of finding the logarithm of a fractional number between 1.0 and 2.0 (the ...
7
votes
4answers
475 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
8answers
200 views

Calculate $\ln 97$ and $\log_{10} 97$

Calculate $\ln 97$ and $\log_{10} 97$ without calculator accurate up to $2$ decimal places. I have rote some value of logs of prime numbers up to $11$. $97$ is a little big. In case it would ...
7
votes
5answers
345 views

How to evaluate $\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$ [duplicate]

This problem appears at the end of Trig substitution section of Calculus by Larson. I tried using trig substitution but it was a bootless attempt $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
7
votes
2answers
9k views

What's the formula to solve summation of logarithms?

this is my first question here. I'm studying summation and everything I know is that: $\sum_{i=1}^n\ k$ is $\frac{n(n+1)}{2}\ $ $\sum_{i=1}^{n}\ k^2$ is $\frac{n(n+1)(2n+1)}{6}\ $ $\sum_{i=1}^{n}\ ...
7
votes
3answers
714 views

What's wrong with this logarithm calculation?

We know that $\displaystyle\log_a(xy)=\log_ax+\log_ay$. Consider the following: $$\displaystyle\ln(1)=\displaystyle\ln((-1)\times(-1))=\displaystyle\ln(-1)+\displaystyle\ln(-1) $$ $\...
7
votes
4answers
2k views

Limit to infinity with natural logarithms $\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $

I found the following problem in my calculus book: Solve: $$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$ I tried to solve it using log rules and l'Hôpital's rule with no ...
7
votes
5answers
309 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. ...
7
votes
1answer
125 views

Is $a^{\ln b} = b^{\ln a}$?

I was struggling with a math problem, namely, a limit with a power to the log of something. While I was struggling with it, I found out that $$a^{\ln b} = b^{\ln a}$$ for all positive values that I've ...
7
votes
4answers
538 views

Doubt about the domain in logarithmic functions.

According to my book, the logarithmic function $$\log_{a}x=y$$ is defined if both $x$ and $a$ are positive and $x\neq 0$ and $a\neq 1$. So are these not correct? $$\log_{-3}9=2$$ $$\log_{-2}-8=3$$
7
votes
2answers
167 views

Solving for $x$ in $3^{2x+1} = 3^x + 24$

I'm having trouble solving this equation step by step: $$3^{2x+1} = 3^x + 24$$ I've tried to take the log of both sides but then I am stuck with the right hand side being $\log(3^x + 24)$. I've ...
7
votes
6answers
363 views

Elegant way to solve $n\log_2(n) \le 10^6$

I'm studying Tomas Cormen Algorithms book and solve tasks listed after each chapter. I'm curious about task 1-1. that is right after Chapter #1. The question is: what is the best way to solve: $n\lg(...
7
votes
3answers
150 views

Finding $\int_0^{\pi/4}\sqrt{1+\left( \tan x\right)^2}dx$

I would like to understand all the steps to find out this integral $$ \int_0^{\pi/4} \sqrt{1+\left( \tan x\right)^2} dx$$ Wolfram Alpha returns: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
7
votes
3answers
483 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
2answers
810 views

Motivation for definition of logarithm in Feynman's Lectures on Physics

I'm not sure if the title is descriptive enough; feel free to change it if you come up with something better. I've been reading through Feynman's Lectures on Physics. In the first volume, he ...
7
votes
2answers
640 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was confused....
7
votes
2answers
126 views

Solving a logarithmic equation $\log_2 (2^x-1)+x=\log_4 (144)$

I need to solve this: $$\log_2 (2^x-1)+x=\log_4 (144)$$ I can work out that $x=\log_2 (2^x)$ and $\log_4 (144)=log_2(12)$ but I'm stuck after that.
7
votes
2answers
146 views

Why is this function a really good asymptotic for $\exp(x)\sqrt{x}$

$$f(x)=\sum_{n=0}^{\infty} a_n x^n\;\;\;\;\; a_n = \frac{1}{\Gamma(n+0.5)}$$ Why is this entire function a really good asymptotic for $\exp(x)\sqrt{x}$, where for large positive numbers, $f(x)\exp(-x)...
7
votes
4answers
315 views

What did Johann Bernoulli wrong in his proof of $\ln z=\ln (-z)$?

Some people say, Johann Bernoulli has proven $\ln z=\ln (-z)$ in the following way $$\ln ((-z)^2 )=\ln(z^2)\;\;\;\Rightarrow\;\;\;2\ln(-z)=2\ln z\;\;\;\Rightarrow\;\;\;\ln (-z)=\ln z$$ While the ...
7
votes
2answers
433 views

Logarithms of logarithms of Graham's number, is the result ever handy?

The other day I was asked how to represent really big numbers. I half-jokingly replied to just take the logarithm repeatedly: $$\log \log \log N$$ makes almost any number $N$ handy. (Assume base 10)....
7
votes
2answers
412 views

Integral $\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$

How can I evaluate following logarithmic integral: $$\int\limits_0^1 \frac{\ln x\ln ( 1 - zx )}{1 - x} dx$$
7
votes
2answers
148 views

Integral $\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$

I need to evaluate this integral: $$\int_0^1\arctan(x)\arctan\left(x\sqrt3\right)\ln(x)dx$$ Apparently, Maple and Mathematica cannot do anything with it, but I saw similar integrals to be evaluated in ...
7
votes
4answers
3k views

Prove that $\log X < X$ for all $X > 0$

I'm working through Data Structures and Algorithm Analysis in C++, 2nd Ed, and problem 1.7 asks us to prove that $\log X < X$ for all $X > 0$. However, unless I'm missing something, this can't ...
7
votes
1answer
113 views

$a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$

Suppose that $a_{n+1}=\log(1+a_n),~a_1>0$. Then find $\lim_{n \rightarrow \infty} n \cdot a_n$. I can find $\lim_{n \rightarrow \infty}a_n=0$. But I have no idea to find $\lim_{n \rightarrow \...
7
votes
1answer
109 views

How to solve $\log _{x^{2}-3}(x^{2}+6x)<\log _{x}(x+2)$?

How to solve the following inequality $$\log _{x^{2}-3}\left(x^{2}+6x\right)<\log _{x}(x+2)\ ?$$
7
votes
1answer
497 views

Are Base Ten Logarithms Relics?

Just interested in your thoughts regarding the contention that the pre-eminence of base ten logarithms is a relic from pre-calculator days. Firstly I understand that finding the (base-10) ...
7
votes
2answers
3k 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} ...
7
votes
5answers
202 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
7
votes
1answer
269 views

ln(z) as antiderivative of 1/z

When integrating $$\frac{1}{x}$$ (where $x \in \mathbb{R} $) one gets $$ln|x|+c$$ since for $x>0$ $$(ln|x|+c)'=(ln(x)+c)'=\frac{1}{x}$$ and for $x<0$ $$(ln|x|+c)'=(ln(-x)+c)'=\frac{1}{-x}(-1)=\...
7
votes
1answer
161 views

Differentiating $y=x^x$ with the formal definition of a derivative

A friend and I were messing around with derivatives, and while we both know the procedure for finding the derivative of $y=x^x$ with logarithmic differentiation, i.e. $$y=x^x\\ ln(y)=x*ln(x)\\ \dfrac{...
7
votes
1answer
2k views

where do exponential and logarithmic functions intersect?

If $0<a<1$, then the graphs of $y=a^x$ and $y=\log_a(x)$ intersect at some point $(t(a),t(a))$. Does this function $t(a)$ have any nice expression? How much do we know about this function, ...
6
votes
7answers
1k views

Prove that $5/2 < e < 3$?

Prove that $$\dfrac{5}{2} < e < 3$$ By the definition of $\log$ and $\exp$, $$1 = \log(e) = \int_1^e \dfrac{1}{t} dt$$ where $e = \exp(1)$. So given that $e$ is unknown, how could I ...
6
votes
5answers
1k views

How do you solve a logarithm with a non-integer base?

How would one calculate the log of a number where the base isn't an integer (in particular, an irrational number)? For example: $$0.5^x = 8 \textrm{ (where } x = -3\textrm{)}$$ $$\log_{0.5}8 = -3$$ ...
6
votes
2answers
446 views

How do I solve for $x$ in $\ln(x)\ln(x) = 2 +\ln(x)$

How do I solve for $x$? $$\ln(x)\ln(x) = 2 +\ln(x)$$
6
votes
10answers
2k views

How can I find the value of $\ln( |x|)$ without using the calculator?

I want to know if there is a way to find for example $\ln(2)$, without using the calculator ? Thanks
6
votes
5answers
1k views

Am I allowed to apply L'Hospital's Rule inside of the natural logarithm function?

I have the following limit: $$\lim_{x\rightarrow \infty} \ln\left(\frac{2x^2+1}{x^2+1}\right)$$ If I was finding the limit of only the terms inside the natural log function, I would have the ...
6
votes
4answers
206 views

Proof of $\log_xy=\frac{\log_zy}{\log_zx}$

Why is $\log_xy=\frac{\log_zy}{\log_zx}$? Can we prove this using the laws of exponents?