Questions related to real and complex logarithms.

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7
votes
3answers
199 views

Integral $\int_0^{1/2}\arcsin x\cdot\ln^2x\,dx$

I'm interested in this integral $$\int_0^{1/2}\arcsin x\cdot\ln^2x\,dx$$ My idea was to first evaluate $$\int_0^{1/2}\arcsin x\cdot ...
4
votes
0answers
78 views

How prove that $\log_3{5} + \log_2{5}$ is irrational? [duplicate]

How prove that: $\log_3{5} + \log_2{5}$ is irrational? $$\log _3{5} + \log _2{5}=\frac{1}{\log _5{3}}+ \frac{1}{\log _5{2}}= \frac{\log _5\left( 3\cdot 2\right) }{\log _53 \cdot \log _52}$$
2
votes
1answer
37 views

Approximating logarithmic series

I'm trying to solve the below recurrence $$T(n) = 2 T(n/2) + n/\log n$$ I referenced some online articles they all are approximating $n \sum_{i=0}^{h-1} \frac{1}{\log n - i}$ to $n ...
0
votes
1answer
38 views

converting the sum of numbers to logarithms

If I have three number, $[0.2, 0.3, 0.4]$ then I can take the sum simply by adding the term: $0.2+0.3+0.4=0.9$. The proportion of the first element is then $0.2/0.9 = 2/9$. Now I don't know what the ...
1
vote
3answers
56 views

Why can't you add a coefficient before the logarithm base change rule?

Why is it that the rule $$ \log_b(x) \ = \ \frac{\log_c(x)}{\log_c(b)} \ $$ (the logarithm base change rule) is true but $$ \ a \log_b(x) \ = \ \frac{a \log_c(x) }{ a \log_c(b) } \ $$ isn't? For ...
2
votes
2answers
139 views

Is $0 \times \ln(0) =\ln(1) $ true?

Can we affirm that: $0 \times \ln(0) = \ln(0^0) = \ln(1) = 0$? The problem is $\ln(0)$ is supposed to be undefined but it works
0
votes
1answer
31 views

Why does the branch cut for log(1+z), cutting away the negative axis, start at -1?

I am really confused about this concept. Is it because from -1 to 0, the inputs (1+z) are just viewed as inputs that aren't actually from the negative axis at all, and so log(1+z) is well-defined ...
1
vote
2answers
32 views

Can the complex square root of $z\sin z$ be defined in a neighborhood of the origin? (I.e., including the origin)

Edit: on a second thought, I don't think it's possible since $$ f(z) = \sqrt {z\sin z} = e^{\large \frac{1}{2} \log z}e^{\large \frac{1}{2} \log\sin z}$$ $$e^{\large \frac{1}{2} (\ln|z| + ...
18
votes
2answers
387 views

Conjecture $\int_0^1\ln\ln\left(\frac{1+x}{1-x}\right)\frac{\ln x}{1-x^2}\,dx\stackrel?=\frac{\pi^2}{24}\,\ln\left(\frac{A^{36}}{16\,\pi^3}\right)$

I did some numeric experiments with integrals involving double logarithms (because they received much interest both on this site and in published papers, sometimes under names of ...
1
vote
1answer
34 views

Determining the value of parameters given constraints

If $$\frac{x(y+z-x)}{\log x}=\frac{y(z+x-y)}{\log y}=\frac{z(x+y-z)}{\log z}$$ and $$ax^yy^x=by^zz^y=cz^xz^y$$ then what is the value of $a + \frac b c$? I am getting as $ax^yy^x=by^zz^y=cz^xx^z$ ...
2
votes
2answers
93 views

Prove $f(x)=2^x$ is continuous

I have show that $f(x)=2^x$ is continous by using the Weierstrass definition (epsilon-delta). I set apart two cases. The first one $x>x_0$ was good. The second one is the problem right now. ...
1
vote
3answers
85 views

The domain of $\ln(x)^{\ln(x)}$

I'm a little bit confused! what is the domain of this function: $$ \ln(x) ^{ \ln(x) } $$ this function, in fact, is: $$ \exp(\ln(\ln(x))\cdot\ln(x)) $$ so the domain would be: $$ x>1 $$ But: ...
1
vote
2answers
40 views

How to use the properties of the logarithmic function

I'm coding the game asteroids. I want to make a levels manager who can create a infinity number of level increasing in difficulty. My levels have as parameters : The number of asteroids on the ...
7
votes
2answers
129 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 ...
3
votes
4answers
87 views

Can you compute $\int_0^1\frac{\log(x)\log(1-x)}{x}dx$ more precisely than $1.20206$ and do a comparision with $\zeta(3)$?

I know from Wolfram Alpha that $$\int_0^1\frac{\log(x)\log(1-x)}{x}dx=1.20206$$ and in the other hand, too from this online tool that ...
0
votes
3answers
39 views

Log transformation

Supose I have a series of numbers from 1 to 10. Their mean value is 5.5. Now supose I apply some transformation like $y=2x+1$. Now their mean value is 12. Now, if I want to get back the original mean, ...
4
votes
1answer
121 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
1
vote
2answers
79 views

How to solve this equation algebraically?

I've come across this interesting equation which I do not know how to solve. The equation is: $$e^x+\log x =1$$ I used WolframAlpha to solve it and it got but, it didn't provide any solutions. The ...
7
votes
1answer
122 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 ...
0
votes
2answers
23 views

Question about expressing logarithms

If logb (a) = m and logy (b) = c, then find loga (y) in terms of variable c and m This is what I have so far logb (a) = log a / log b logy (b) = log b / log y =(log a / log b)(log b / log y) = ...
0
votes
3answers
70 views

Logarithm of a transcendental number

Can anything be said about the nature of the number $\log y $ where $y $ is a transcendental number not of the form $y=e^x $ or written trivially in that form using $x=\log w $ for some $w $ ...
4
votes
2answers
103 views

Can $\log(x)\log(y)$ be reduced?

I'm currently taking Pre-Calc and am learning about logs. I know that $\log(xy) = \log(x) + \log(y)$, but can $\log(x)\log(y)$ be reduced further?
1
vote
0answers
14 views

Multiplicative & additive measurement error models concerning logarithms

I understand that taking the logarithm of the multiplicative error model transforms it into the additive error model. Let $y'$ be the observed response variable, with $y$ being the true response ...
1
vote
2answers
58 views

If $\ln x$ is integrable, then is $x \ln x$ also integrable?

I have a very simple problem. Assume we have a finite measure $\mu$ on $[1,\infty)$, and \begin{align} \int_1^\infty t ~d\mu(t) < \infty. \end{align} My question is if this implies \begin{align} ...
-1
votes
2answers
38 views

Simplifying a log function

How come: y=c1*e^(c2*t) is simplified to: ln(y)=ln(c1)+c2*t ? What I got is: ...
4
votes
0answers
79 views
7
votes
1answer
67 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 ...
0
votes
1answer
16 views

Inequality for Difference Between Two Harmonic Numbers

I want to know if the following inequality holds: $H_n-H_k \leq \log{n}-\log{k}$ where $k \leq n$ and $n \to \infty$. I have seen two similar questions such as this question and this one but I could ...
5
votes
2answers
86 views

Finding the logarithm of a matrix?

Find $B$ if $A=e^B$ and $A=\begin{bmatrix} 2&1&0\\ 0&2&0\\ 0&0&4\\ \end{bmatrix}$. Besides, I would be very happy if give some general remark(Best approach). I have seen the ...
5
votes
3answers
94 views

Inequality $\log x\le \frac{2}{e} \, \sqrt{x}$

The inequality $$\log x \le \frac{2}{e} \, \sqrt{x},$$ where $\log x$ denotes the natural logarithm, is used in the proof of Theorem 4.7 in Apostol's Analytic Number Theory. It seems that the ...
0
votes
1answer
24 views

Solving for n. Logarithm [closed]

How come: $\dfrac{3^n}{2^{2n-1}}<0.5\cdot10^{-5}$ will be equal to $n = 45$?
1
vote
2answers
59 views

Calculus - limit of a function with logarithms

Compute $$\lim_{x \to {0_+}} {\ln (x \ln a) \ln \left({{\ln (ax)}\over{\ln({x\over a})}}\right)}$$ where $a>1$ I am trying to get to a result withou using any advanced methods or even things such ...
0
votes
1answer
191 views

Derivatives and Integrals of Polynomials and more.

I noticed that if I had a function $f(x)=x^n$ where $n$ is an integer, then $\lim_{m\to{n^+}}f^{(m)}(x)=n!$ where $f^{(m)}(x)$ is the $m$-th derivative. Also, ...
1
vote
2answers
60 views

Unitary matrix in trace and log function

I am trying to do a unitary transformation $U$ on square matrix $A$ which is embedded inside a trace and natural log function, and the following property is supposed to hold: $\mathrm{tr} (\ln (A)) = ...
5
votes
4answers
71 views

Solve the equation: $2^{2x+1}=\left(\frac{1}{32}\right)^x$

Having trouble with this problem: $$2^{2x+1}=\frac{1}{32^x}$$ Do I need to set the exponents equal to each other?
0
votes
1answer
21 views

Logarithm rule for branch cut logarithms

I know that for $a, b \in \mathbb{R}$, the rule $\log(ab) = \log(a) + \log(b)$ holds. What about for $a_1, b_1$ in the right half-plane, or $a_2, b_2$ in the sector from $\frac{-3\pi}{4}$ to ...
0
votes
0answers
14 views

How do you solve the following (logs)

how do you solve the following? My intuition tells me to take exp and then raise it to five to get a polynomial. But, it stops there. Any ideas? $0.8 \log (9-0.2x) + 0.2 \log (2+0.8x) = 0.6 ...
1
vote
1answer
49 views

Positive logarithm in a $C^*$-algebra

Let $A$ be a $C^*$-algebra and $a \in A_+$ be a positive element. I want to show that $a$ has a positive logarithm if $a$ is invertible. I just see that the usual $\log$ function is continuous on ...
1
vote
2answers
42 views

Prove that $\log{\frac{\sum_{j=1}^n x_j}{n}}\ge \frac{\sum_{j=1}^n \log x_j}n$ for $j=1,…,n$

Prove that $\log{\frac{\sum_{j=1}^n x_j}{n}}\ge \frac{\sum_{j=1}^n \log x_j}n$ for $j=1,...,n$. My Work. LHS: \begin{align}\log{\frac{\sum_{j=1}^n x_j}{n}}&=\log{\sum_{j=1}^n x_j}-\log ...
0
votes
4answers
79 views

Why is this proof considered wrong? [closed]

I was asked to prove following statement: $$\log_a{(x_1\cdot x_2)} = \log_a{(x_1)} + \log_a{(x_2)}$$ What I did was: \begin{align} \log_a{(x_1\cdot x_2)} &= \log_a{(x_1)} - (-1)\cdot ...
0
votes
1answer
46 views

How to solve $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$?

Let k be positive integers $\lim_{x\rightarrow \infty }\dfrac {2} {x}\sum _{k\rightarrow 1}^{x}\ln \left( \dfrac {x+k} {x}\right)$
0
votes
2answers
24 views

In Need of Logarithms Simplification Exercises

I am very interested in mathematics, however, finding nowhere near wanted information in school sometimes I go and learn something by myself. Just like this time. I decided to learn more about ...
0
votes
1answer
40 views

Find the values of $x$ satisfying the equation $ { \log_{5} }^2 x + \log_{5x} {5\over x } = 1 $?

$$ { \log_{5} }^2 x + \log_{5x} {5\over x } = 1 $$ My progress $$\begin{align} &{ \log_{5} }^2 x + \log_{5x} {5\over x } = 1 \\[2ex] \implies & { \log_{5} }^2 x + \log_{5x} {25} \cdot ...
1
vote
6answers
37 views

Logs with exponential bases

I know that $e^{\log_{e^2} (16)}$ is 4, but I can only get this far: $$e^{\log_{e^2}(4^2)}$$ $$e^{2\log_{e^2}(4)}$$ I need some way to cancel the 2's so I can get $$e^{\log_e(4)}$$ but I don't the ...
0
votes
2answers
27 views

Solve for $x$ in an system of a linear and logarithmic equation.

I have this question on my homework: $$\\f(x)=\ln{(x-3)}\\g(x)=\frac12x-7\\\text{solve for x in:}f(x)=g(x)$$ I have used the substitution property to get this: $$\ln{(x-3)}=\frac12x-7$$. I don't ...
0
votes
0answers
45 views

Solving for $x$ in $(y-x)\ln\frac{x}{y} = a$

I have the expression $$(y-x)\ln\frac{x}{y} = a,$$ and I want to express $x$ in terms on $y$ and $a$. I know that in this kind of problem, the Lambert function $W$ is likely to show-up, but that ...
0
votes
2answers
27 views

Need help with expressing this logarithm

Express $log_3(a^2 + \sqrt{b})$ in terms of m and k where $m = log_{3}a$ $k = log_{3}b$ Given this information I made $a = 3^m$ $b = 3^k$ Therefore = $log_{3} ((3^m)^2 + (3^k))^{\frac{1}{2}}$ = ...
0
votes
3answers
33 views

Solving a log equation for two variables

Goal is to find both $\beta$ and $\omega$. I already have the answer here, but I'm confused as to how to get it. $\log_6 250 - \log_\beta 2 = 3 \log_\beta \omega$ This is what I did: $\log_6 250 = ...
2
votes
0answers
46 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from ...
1
vote
2answers
21 views

Unsure how to treat y in this derivative/log problem

Need to find the derivative of $h(y)= \ln(y^2 \cos y)$ Treating it like a normal variable like an x isn't working for me, the way we used y's in earlier problems where you get a y' in there doesn't ...