Questions related to real and complex logarithms.

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6
votes
1answer
102 views

Complex exponent properties?

Here is a line in a proof in a complex analysis text: $\sqrt{1-z^2}=\sqrt{1-z}\sqrt{1+z}$ I know you can't do this in general, but when can you do it? Here is what I tried: ...
6
votes
1answer
251 views

Complex Logs and Roots of Unity

I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...
6
votes
0answers
219 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
6
votes
1answer
127 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)\\ ...
5
votes
6answers
897 views

Alternate proof for “$\log_{10}{2}$ is irrational”

I need to prove that $\log_{10}{2}$ is irrational. I understand the way this proof was done using contradiction to show that the even LHS does not equal the odd RHS, but I did it a different way and ...
5
votes
7answers
724 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 ...
5
votes
2answers
105 views

Evaluate $\lim\limits_{y\to 0}\log(1+y)/y$ without LHR or Taylor series

Problem 12 on p. 297 of Spivak's Calculus (first edition) is Find $\lim\limits_{y\to 0}\log (1+y)/y$. (You can use L'Hospital's rule, but that would be silly.) I'm not sure the other method ...
5
votes
7answers
2k views

Is there any significance to the logarithm of a sum?

Many years ago, while working as a computer programmer, I tracked down a subtle bug in the software that we were using. Management had dispaired of finding the bug, but I pursued it in odd moments ...
5
votes
5answers
220 views

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

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 $$\int_0^1 \frac{\ln(x+1)}{x^2+1} dx$$
5
votes
6answers
245 views

Prove that for all $x>0$, $1+2\ln x\leq x^2$

Prove that for all $x>0$, $$1+2\ln x\leq x^2$$ How can one prove that?
5
votes
4answers
129 views

Can someone explain why $x^{\log(a)} = a^{\log(x)}$?

I'm trying to see why the below is true. $$ x^{\log(a)} = a^{\log(x)} $$ Anyone here know why this is? Thank you.
5
votes
5answers
435 views

L'Hôpital's as $x$ tends to infinity

I'm searching for the explanation to the limit of: $$ \lim\limits_{x\to\infty} x\, \ln\frac{x+1}{x-1}. $$ I know the answer is 2, but I can't seem to get there. The problem is in my textbook under a ...
5
votes
5answers
532 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 ...
5
votes
3answers
384 views

Solving exponential equations using logarithms

This is the equation that I am having troubles with: $$\large x^{\large\log_{10}5}+5^{\large\log_{10}x}=50$$ So the first thing I do, I logarithm the whole expression with $\log_{10}$. So I get: ...
5
votes
3answers
241 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 ...
5
votes
4answers
242 views

Mathematical notation/name for the number of times a number can be divided by 2

I am using this simple snippet of code, variants of which I have seen in many places: for(int k = 0 ; n % 2 == 0 ; k++) n = n / 2; This code repeatedly ...
5
votes
2answers
465 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
5
votes
4answers
166 views

Is $\ln\sqrt{2}$ irrational?

I know that the natural log of any positive algebraic number is transcendental, as a consequence of the Lindemann-Weierstrass theorem, but what about the natural log of the square root of two (which ...
5
votes
5answers
142 views

Showing that $e^{-2} < \ln 2$

I have to prove the following inequality: $e^{-2} < \ln2.$ Using Bernoulli's inequality, I showed that $2 \leq e$, and using this result I tried to simplify the inequality by using an upper ...
5
votes
2answers
215 views

Solving $x^2 - 1 = e^x$

Can someone help me solve the equation $x^2 - 1 = e^x$ ? I tried taking the natural logarithm of both sides but I don't know where to go from there.. I got: $\ln(x^2 -1) = x$ But I don't know how ...
5
votes
3answers
723 views

Verifing $\int_0^{\pi}x\ln(\sin x)\,dx=-\ln(2){\pi}^2/2$

I used all I know to show that $$\int_0^\pi x\ln(\sin x)dx=-\ln(2) \pi^2/2$$ This is my homework but don't know where to start. I appreciate your help.
5
votes
2answers
2k 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
4answers
146 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
5
votes
3answers
822 views

How to solve $n < 2^{n/8}$ for $n$?

This is from an exercise (1.2.2) in introduction to algorithms that I'm working on privately. To find at what point a $n \lg n$ function will run faster than a $n^2$ function I need to figure out for ...
5
votes
4answers
1k views

Summation of logs

Are there any useful identities for quickly calculating the sum of consecutive logs? For example $\sum_{k=1}^{N} log(k)$ or something to this effect. I should add that I am writing code to do this (as ...
5
votes
3answers
203 views

A little question about logarithm

Can we say that $\log _4 (n^2)=\log _2(n)$? If that is the case, then $\displaystyle 2^{\log _4 (n^2)}=n$? Thanks.
5
votes
4answers
325 views

Can we prove $a^{\log_bn} = n^{\log_ba}$?

Can we prove $$a^{\log_bn} = n^{\log_ba}?$$ I forget how to prove this theorem. I picked up one numbers for test, and they worked.
5
votes
3answers
263 views

Solving the equation $3^{5x-2}=8^{8x-9}$

I'm trying to solve the equation $$3^{5x-2}=8^{8x-9}.$$ I'm assuming I need to do some work with logarithms, but I don't know what to do. Thanks in advance!
5
votes
3answers
181 views

Find the value of $x$ such that $2^x=10$

Given that $\log 5 = 0.7$ (to one decimal place), find the value of $x$ such that $2^x = 10$ (again to one decimal place) I don't know what to do with the information that $10^{0.7} = 5$. Why is ...
5
votes
1answer
162 views

Why does $\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$

I read on Wikipedia that $$\cos (\pi\cos (\pi \cos (\log (20+\pi)))) \approx -1$$ to a high degree of accuracy. Why is this true? Is this pure coincidence or is there some mathematical ...
5
votes
2answers
225 views

Homework with logarithms

I'm stuck on continuing the next exercise: Considering: $\log_{c}a = 3$ $\log_{c}b = 4$ and: $$ y = \frac{a^{3}\sqrt{b \cdot c^{2}}}{2} $$ What's the value of $\log_{c}y$ ...
5
votes
2answers
184 views

Derivative of ${ x }^{ x }$ without logarithmic differentiation

With logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} ...
5
votes
3answers
286 views

Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?

I think this might be a case of slight ambiguity in notation, but here goes: On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
5
votes
3answers
80 views

Equation with Logarithm in Exponent

How to I solve the following exercise with a logarithm? I've forgotten the "trick" for doing so: $x^{log_{10} x} =10^4$
5
votes
4answers
202 views

To find the logarithm of $1728$ to the base $2 \sqrt{3}$

Find the logarithm of: $1728$ to base $2\sqrt{3}$. Let, $\log_{2\sqrt{3}} 1728 = y$, then $$\begin{align} (2\sqrt{3})^y &= 1728\\ 2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
5
votes
4answers
11k views

Units of a log of a physical quantity

So I have never actually found a good answer or even a good resource which discusses this so I appeal to experts here at stack exchange because this problem came up again today. What happens to the ...
5
votes
3answers
186 views

Summation of a series.

I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series : ...
5
votes
4answers
351 views

What are logarithms?

I have heard of logarithms, and done very little research at all. From that little bit of research I found out its in algebra 2. Sadly to say, I'm going into 9th grade, but yet I'm learning ...
5
votes
4answers
112 views

How is $\ln(-1) = i\pi$?

How do I derive: $\ln(-1)=i\pi$ and $\ln(-x)=\ln(x)+i\pi$ for $x>0$ and $x \in\mathbb R$ Thanks for any and all help!
5
votes
1answer
148 views

Prove this polylogarithmic integral has the stated closed form value

Question. Prove the following polylogarithmic integral has the stated value: $$I:=\int_{0}^{1}\frac{\operatorname{Li}_2{(1-x)}\log^2{(1-x)}}{x}\mathrm{d}x=-11\zeta{(5)}+6\zeta{(3)}\zeta{(2)}.$$ ...
5
votes
3answers
72 views

Why does $\int^{ab}_{a} \frac{1}{x} dx = \int^{b}_{1} \frac{1}{t} dt$?

I can't understand how the integral having limits from $a$ to $ab$ in Step 1 is equivalent to the integral having limits from $1$ to $b$. I'm a beginner here. Please explain in detail. ...
5
votes
1answer
469 views

Why does the method to find out log and cube roots work?

To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths. 1)Take the number $X$, whose cube root needs ...
5
votes
4answers
3k views

How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
5
votes
1answer
1k views

$\log_7 n$ is either an integer or an irrational number

Show that $\log_7 n$ is either an integer or an irrational number where n is a positive number. I assumed that it is rational and tried to get a contradiction for $\log_7 n = a/b$, where b does ...
5
votes
2answers
513 views

continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
5
votes
3answers
337 views

Solving $\log _2(x-4) + \log _2(x+2) = 4$

Here is how I have worked it out so far: $\log _2(x-4)+\log(x+2)=4$ $\log _2((x-4)(x+2)) = 4$ $(x-4)(x+2)=2^4$ $(x-4)(x+2)=16$ How do I proceed from here? $x^2+2x-8 = 16$ $x^2+2x = 24$ ...
5
votes
3answers
97 views

What's a good class of functions for bounding/comparing ratios of complicated logarithms?

I have this goofy series $\sum \limits_{n=2}^\infty \frac{ \log_2 \left[ n \log_2^2 n \right]}{n \log_2^2 n}$ that Wolfram Alpha tells me diverges by the comparison test (and indeed, in the larger ...
5
votes
2answers
420 views

Prove that $\log(1 + \sqrt{1+x^2})$ is uniformly continuous?

Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous. My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = ...
5
votes
2answers
261 views

How can I solve $x^x = 5$ for $x$? [duplicate]

Possible Duplicate: Is $x^x=y$ solvable for $x$? I've been playing with this equation for a while now and can't figure out how to isolate $x$. I've gotten to $x \ln x = \ln 5$, which seems ...
5
votes
1answer
2k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...