Questions related to real and complex logarithms.

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6
votes
0answers
175 views

Integral ${\large\int}_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx$

Could you please help me to find closed form expressions for the following definite integrals: $$I_1=\int_0^{\pi/2}\frac{x\,\log\tan x}{\sin x}\,dx\approx0.3606065973884796896...$$ $$I_2=\int_0^{\pi/3}...
2
votes
1answer
31 views

How to rigorously deduce the Laurent series of $\log\frac{z-p}{z-q}$?

Of course, the logarithm here is defined on the ring region $|z|>R\ge\max\{|p|,|q|\}$ as $$\log\frac{z-p}{z-q}=\int_{z_0}^z \left(\frac1{w-p}-\frac1{w-q}\right)\mathrm d w. $$ Here the integral is ...
0
votes
2answers
57 views

Logarithmic equation's solution [closed]

I'm unable to solve this equation further. Could someone have a shot at it and try to solve it and explain it to me please? The equation is $$2\log_{2}(\log_{2}(x))-\log_{2}(\log_{2}2\cdot 2^{0.5}x)=...
-2
votes
1answer
58 views

What would $\log_{24} 48$ be in terms of $x$ if $\log_{12} 36= x$? [closed]

What would $\log_{24} 48$ be in terms of $x$ if $\log_{12} 36= x$? Which properties to use and how to proceed, please let me know.
2
votes
2answers
52 views

For which values of $a$ does $d\ge ac\ln c\implies d\ge c\ln d$?

For which values of $a>0$ is it true that for all $c,d>0$, $\hspace{.2 in}d\ge ac\ln c\implies d\ge c\ln d$? I believe that this is true for $a\ge2$, (see Showing if $n \ge 2c\log(c)$ then $n\...
0
votes
1answer
63 views

Problem on logarithm

$$\mbox{If}\ \log_{2a}\left(a\right) = x,\ \log_{3a}\left(2a\right) = y,\ \log_{4a}\left(3a\right) = z.\quad \mbox{Then, what is the value of}\ xyz-2yz\,?. $$ Not exactly able to solve it any further.
0
votes
2answers
83 views

Logarithmic equation solution

If $ \frac {\log a}{b-c}=\frac{\log b}{c-a}=\frac{\log c}{a-b}$, then what would be the value of $a^{b+c}.b^{c+a}.c^{a+b}$? I'm unable to proceed.
16
votes
1answer
348 views

Convexity of difference of log-sum-exp: $f(x_1, x_2, x_3, x_4) = \log(e^{x_1} + e^{x_2}) - \log(e^{x_1} + e^{x_2} + e^{x_3} + e^{x_4})$

I would like to know whether the following function $f: \mathbf{R}^4 \to \mathbf{R}$ is concave or not: $$ f(x_1, x_2, x_3, x_4) = \log(e^{x_1} + e^{x_2}) - \log(e^{x_1} + e^{x_2} + e^{x_3} + e^{x_4})...
0
votes
3answers
53 views

question from logarithms [closed]

Simplify without using tables $$\frac{\log25+\log625}{\log5}$$
1
vote
2answers
126 views

$\log(e^z - i)$ as a holomorphic function in $\mathbb{D}$

I'm learning complex analysis, specifically holomorphic functions, and need help with the following exercise: Examine if the function $\log(e^z - i)$ can be defined as a holomorphic function in ...
3
votes
4answers
53 views

Find $\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$

I have to solve this limit: $$\lim_{n\to\infty}\ln(n^{n}\cdot(n+1)^{-n-1})$$ I know the answer is $-\infty$. My question is, can I do this: $$\ln[\lim_{n\to\infty}n^{n}\cdot(n+1)^{-n-1}]$$ If not, how ...
0
votes
1answer
76 views

$\log_5 {(x+1)} - \log _4 {(x-2)} = 1$

I tried to solve this equation by changing bases. $\dfrac {\log_4 (x+1)}{\log_4 5} = \log_4 (4x -8)$ $.86 \log_4 (x+1) = \log_4 (4x-8)$ Then i got stuck. Please share your idea with me.
1
vote
2answers
39 views

Defining logarithms explicitly

How are logarithms defined explicitly? One way I can come up with is the following, first start by using McLaurin series representation of exponent function: $$e^x = \sum_{k=o}^{\infty}\frac{x^k}{k!}...
0
votes
2answers
53 views

Calculate $\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$

I have this exercise:$$\log_{\frac{1}{2}}{(2\sqrt[3]{4})}$$ There is not equation or something else, how should I solve this type of exercise?
0
votes
2answers
53 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
2
votes
2answers
35 views

Solve the following (logarithmic) function for x

$x^{log_{2}x}+16x^{-log_{2}x} = 17$ Looks horrible, I started by removing the exponents: $e^{ln(x)*log_{2}x}+16e^{-ln(x)*log_{2}x}=17$ | ln() $ln(x)*log_{2}x-16ln(x)*log_{2}x=ln(17)$ $ln(x)*log_{...
2
votes
2answers
43 views

Solve the following (logarithmic) function for $x$

$(\log_{3}x)^{2} - 3\log_{3}x + 2 = 0$ We may not use many rules, so I would start by ignoring the ^(2), ignore -3* but take ...
1
vote
0answers
7 views

Caclulate Standard Error when subtracting two effects

I am working on this issue that compares two regions with the McCrary Test (Regression Discontinuity Design). It basically measures how the density of a variable of interest develops accross a pre set ...
1
vote
1answer
34 views

Question about distribution of primes

The following is from "Introduction to Number Theory" by Hardy and Wright. The book first states the following theorem Theorem A: If $\pi(x)$ is number of primes not exceeding $x$ then $\pi(x) \...
5
votes
2answers
98 views

Finding Limit of Nested/Continued Logarithm

For a sequence $a_n$ defined by: $$a_1 = \ln(1)$$ $$a_2 = \ln\left(\frac{1}{\ln(2)}+1\right)$$ $$\dots a_n = \ln\left(\frac{1}{\ln(\frac{1}{\ln(\dots 1/\ln(n ))}+1)}+1 \right)$$ with $n$ ...
1
vote
0answers
54 views

If the last term of $(2^{1/3}-\frac{1}{\sqrt{2}})^n$ is $(\frac{1}{3^{5/3}})^{\log(\frac{8}{3})}$, what is the $5\rm{th}$ term from the beginning? [closed]

If the last term of $$\left(2^{1/3}-\frac{1}{\sqrt{2}}\right)^n$$ is $$\left(\frac{1}{3^{5/3}}\right)^{\log(\frac{8}{3})}$$ then the value of $5th$ term from beginning is ?. So I simplified $(243)^{1/...
3
votes
1answer
39 views

How can I prove this inequality involving logarithm?

$n^2 \geq n \log_{2}n$ I tried like this: $n^2 \geq n \log_{2}n$ $n^2-n \log_{2}n \geq 0$ $n(n-\log_2 n) \geq 0$ I don't know what to do after this?
2
votes
2answers
179 views

Formula for proportion of entropy

Let's say we have a probability distribution having 20 distinct outcomes. Then for that distribution the entropy is calculated is $2.5$ while the maximal possible entropy here is then of course $-\ln(\...
2
votes
4answers
98 views

If $\ln(1+x) \approx A+Bx+Cx^2$, differentiate twice both sides and show that $\ln(1+x) \approx x-\frac{1}{2}x^2$

Question: $\ln(1+x) \approx A+Bx+Cx^2$, for $-1<x\leq1$, where $A,B,C$ are constants. Differentiate twice both sides of the approximation above and hence show that $$ \ln(1+x) \approx x-\...
0
votes
2answers
43 views

Basic Logarithms problems [duplicate]

Can someone tell me how many digits would be there in- $(2.5)^{200}$ and $6^{50}? $ I'm utterly confused where to begin from. Any help would be appreciable.
0
votes
1answer
22 views

Characteristics and Mantessa

I've just heard about these terms. Could someone elaborate on what's their use is? And plus could you explain it using a few examples?
0
votes
1answer
35 views

Write $b$ in terms of $a$.

If $a$ and $b$ are both positive and unequal, and: $$\log_ab+\log_ba^{2}=3$$ Find $b$ in terms of $a$. Tidying up a bit, letting; $y=\log_ab$ ; and then solving the quadratic gives two solutions. $$\...
3
votes
3answers
52 views

If $a-c = 9$ then find the value of $b-d$.

If $a,b,c,d$ are positive integers such that $\log_a b=\frac{3}{2}$ and $\log_c d=\frac{5}{4}$, if $a-c = 9$ then find the value of $b-d$. We get $b=a^{3/2}$ and $d=c^{5/4}$ Hence $b-d=a^{3/2}-c^{...
4
votes
5answers
128 views

$\lim_{x\to \infty} \ln x=\infty$

I'm reading the following reasoning: Since $\underset{n\to \infty}{\lim}\ln 2^n=\underset{n \to \infty}{\lim}n\cdot(\ln 2)=\infty$ then necessarily $\underset{x\to \infty}{\lim}\ln x =\infty$. ...
0
votes
4answers
48 views

Solve for $x$: $14^{4x} = 3^{-x-3}$ Write the exact answer using base-10 logarithms

I am having trouble with understanding this question and would appreciate some help and guidance. Solve for x. $$14^{4x} = 3^{-x-3}$$ Write the exact answer using base-10 logarithms
0
votes
1answer
34 views

Solution of given equations for $x$ and $y$

Solve for $x$ and $y$ $$(2x)^{\log 2}=(3y)^{\log 3}$$ $$3^{\log x}=2^{\log y}$$ Could someone give some hint to approach this question?
11
votes
1answer
184 views

Closed form of infinite product $\prod\limits_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$

I encountered this infinite product while solving another problem: $$P(x)=\prod_{k=0}^\infty 2 \left(1-\frac{x^{1/2^{k+1}}}{1+x^{1/2^{k}}} \right)$$ $$P(x)=P \left( \frac{1}{x} \right)$$ I strongly ...
1
vote
3answers
62 views

Solve $3^{x+2}\cdot4^{-(x+3)}+3^{x+4}\cdot4^{-(x+3)} = \frac{40}{9}$

Can someone point me in the right direction how to solve this? $3^{x+2}\cdot4^{-(x+3)}+3^{x+4}\cdot4^{-(x+3)} = \frac{40}{9}$ I guess I have to get to logarithms of the same base. But how? What ...
2
votes
2answers
41 views

Logarithm of >2 numbers

I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected. ...
0
votes
2answers
108 views

Is is true that $\lim(\ln x) = \ln(\lim x)$? [closed]

As the title says, I want to ask everyone if $\lim(\ln x) = \ln(\lim x)$ when x approach to infinite with any function.
0
votes
2answers
19 views

Determine the points where polynomial function intersects logarithmic function

Given the function $n^k$ where $k$ is a constant such that $0<k \leq k_{max}$ where $k_{max}$ is the point at which $n^k$ first intersects with $log_2n$ determine: $k_{max}$ For a given $k$, the ...
3
votes
3answers
22 views

What is the logarithm of $(a-b)\delta_{ij}+b$?

Just now I came across the expression similar to: $x_{ij} = (a-b)\delta_{ij}+b$ The author then somehow converts this expression, into: $\ln x_{ij} = (\ln a-\ln b)\delta_{ij}+\ln b$ This comes ...
2
votes
1answer
63 views

Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n ...
0
votes
0answers
23 views

discrete logarithm with complex numbers

let $z = a + bi$ where $a,b$ are integers on $[0,N)$ let $a + bi \mod t = (a \mod t) + (b \mod t) \cdot i$ Consider the problem of finding $e$ where $z^e \mod N = c$ and $c, N$ and $z$ are known. Is ...
0
votes
0answers
18 views

Finding a branch of the complex logarithmic function $\log(1-z).$

I have a question that asks me to find the holomorphic branch $L(1 − z)$ of $\log(1 − z)$ valid in the cut-plane $z \in \mathbb{C}\setminus [1, ∞)$ and such that $L(1) = 0.$ We have defined the ...
0
votes
0answers
14 views

Infinite exponential sum doubt

Hello! I have a couple of doubts regarding a formula seen here : $$\sum _{k=1}^{\infty } \frac {e^{kz}}{k}= -\log (1-e^{z}) /; Re(z)<0$$ What would happen if the real part of z Re(z) were equal ...
-1
votes
3answers
67 views

Is the following is true? If that so, give me a proof. $-log(1-x)=log(1+e^x)$??

Is the following is true? If that so, give me a proof. $$-log(1-x)=log(1+e^x)?$$ Give me some value where this equality holds. I dont think so it will be same. Because, $$(1-x)^{-1}=1+x+x^2+x^3+\...
1
vote
2answers
57 views

Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$

The question is: Find all solutions of $\left[\ln(\sin^{-1}(e^x))\right]^5=\ln(\sin^{-1}(e^x))$, where $x$ is real. Give the solutions in exact form. What I have done $$\left[\ln(\sin^{-...
2
votes
4answers
78 views

$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$

In wikipedia it says, $$\arctan x=\frac{1}{2}i[\ln(1-ix)-\ln(1+ix)]$$ I want to now why is this true and what does a logarithm of a complex number even mean. I'm guessing that if I use the Taylor ...
0
votes
0answers
48 views

Interpreting $\log_2\left(\frac{1}{0}\right)$ and $\log_2\left(\frac{0}{0}\right)$

I'm hoping someone can confirm what I've done is correct. I am working with biological datasets ... about 100 RNA-Seq datasets and I'm trying to analyze the relative up or down regulation of genes. ...
1
vote
0answers
73 views

Maclaurin Expansion of $\ln(3+x)$

I'm currently evaluating a simple Maclaurin expansion, the confusion I have with is why the expansion of this function is constructed to be: $\ln\left[3\left(1+\dfrac{x}{3}\right)\right]$ as opposed ...
6
votes
4answers
174 views

What are some methods to show $\log$ is not a rational function?

It is easy to show $\log$ isn't a polynomial (no continuous extension to $\mathbb{R}$). More challenging is showing it isn't rational. Suppose it were a rational function. Then write, the fraction ...
3
votes
2answers
49 views

How do I simplify this Log with a Fraction in it?

So I have: $$ \log_2(5x) + \log_2 3 + \frac{\log_2 10}{2} $$ I understand that when there is addition, and the bases are the same, I can simply multiply what is in the parenthesis. So for the first ...
0
votes
1answer
34 views

Find how many years must elapse before the proportions of red kangaroos and grey kangaroos are reversed, assuming the same rates continue to apply.

I have this question (sorry I'm not able to embed it): Q.7. There are approximately ten times as many red kangaroos as grey kangaroos in a certain area. If the population of grey kangaroos ...
0
votes
1answer
35 views

Codomain of p-adic logarithm

We have the natural map $$\log: \mathbb{C}^\times \to \mathbb{R}$$ $$z \to \log |z|$$ Is there a p-adic analogue of this? By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, ...