Questions related to real and complex logarithms.

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0
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0answers
7 views

MSE in case of log-transformed dependent variable

Let's consider the following log-linear model: $log(Y_i) = \alpha + X_i\beta + \epsilon_i$ for i = 1, ..., N The fitted value is: $\widehat{log(Y)} = \hat{\alpha} + X\hat{\beta}$ Assuming ...
0
votes
2answers
23 views

Logarithmic inequality: $\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$

I need help solving this: $$\log_{1/3}^2(x^2-3x+2) - \log_{1/3}(x-1)>\log_{1/3}(x-2) +6$$ So far I could not make sense of this, because I don't understand how to handle $\log^2$ or the $+6$ at ...
0
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2answers
18 views

Express the given expression as a single logarithm

Express $$2 \ln (2 - x) + 3 \ln (x^2 - 5)$$ as a single logarithm. Can anyone help me with this question? Thanks
1
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2answers
28 views

Find $\log_c{x}$ if $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$.

Given that $\log_a{x} = p$, $\log_b{x} = q$, and $\log_{abc} {x} = r$, find the value of $\log_c{x}$.
2
votes
1answer
57 views

Does $\log_a b = \log_\sqrt a \sqrt b$ can be a basic logarithm law?

Does the following equation is true for all $ a,b\in{\mathbb R}$? $$\log_a b = \log_\sqrt a \sqrt b$$ I have tried to proove this, and I didnt find any contradiction. Is it true? EDIT Thanks guys ...
1
vote
1answer
41 views

Maximal number of colours in a palette that allows for fewer than 500 mixtures

An artist is planning on mixing together any number of different colours from her palette. A mixture results as long as the artist combines at least two colours. If the number of possible mixtures is ...
0
votes
2answers
17 views

comparison test to show that $\sum_{n=1}^{\infty}\frac{1}{(n+2)\sqrt{ \ln ^3(n+3)}}$ converges

As the title says, I know that this sum converges and I want to find a suitable comparison test. Cauchy's root test and d'Alembert's ratio test gave inconclusive results. According to wolfram this ...
0
votes
1answer
49 views

Indefinite integrals with natural logs [duplicate]

I know the integral of $\frac{1}{x}$ is $\log(x)$ but I'm not sure how to solve this problem, any help would be appreciated: $$ \int^{3}_{2} \frac{1}{x \ln x} $$ I think I need to substitute $x\ln x$ ...
0
votes
1answer
36 views
2
votes
2answers
41 views

What is the limit of $\lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $

How do i calculate the limit of this function? $$ \lim_{x\rightarrow 0} (\log _{\tan^2x}\tan^22x) $$ I have no idea where to start.
1
vote
7answers
67 views

Calculate $\log_{a}(ab)$, when $\log_{ab}(b)$ is known.

If you know that $\log_{ab}(b) = k$, calculate $\log_{a}(ab)$. Last time I was asked two times about this problem. $a,b$ was given, constant, such that $a,b \in \mathbb{Z} \wedge a,b > 1 ...
0
votes
1answer
75 views

how can I publish my log approximation formula

I've successfully found out a formula which can give log value of any base till 4-5 places after decimal I want to know whether it can get published because I've seen some journals which have ...
6
votes
2answers
122 views

Why does my professor say that writing $\int \frac 1x \mathrm{d}x = \ln|x| + C$ is wrong?

My professor says that writing this is convenient $$\int \frac 1x \mathrm{d}x = \ln|x| + C\tag{1}$$ but wrong, since it should be written as: $$\int \frac 1x \mathrm{d}x = \begin{cases}\ln x + C ...
1
vote
0answers
17 views

Math equation problem - fitting wallpapers on a wall

I am building up a java program but don't have the right idea on how to resolve its math problem. The tasks I am doing are: I have to cover the wall with wallpaper. The wall is "a"(input) meters ...
1
vote
1answer
34 views

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$

Is there any holomorphic function in a unit ball such that $f(1/n)=\frac{1}{n \ln{n}}$ for $n=2,3,\dots$ Can somebody point me in the right direction? Only thing I know is that $f(0)=0$ if such ...
9
votes
4answers
246 views

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

Evaluate $$ \int_{0}^{1} \arctan^{2}\left(\, x\,\right) \ln\left(\, 1 + x^{2} \over 2x^{2}\,\right)\,{\rm d}x $$ I substituted $x \equiv \tan\left(\,\theta\,\right)$ and got $$ ...
0
votes
1answer
13 views

$e^z=-3i$ find $z\in \mathbb C$ check my answer

I am unsure of my solution to this question, since the definition of the complex logarithm is somewhat complex. Since $-3i = 3e^{i\frac{3}{2}\pi}$ we get that $e^z=3e^{i\frac{3}{2}\pi}$ So if we use ...
0
votes
0answers
24 views

complex logarithms

Using complex logarithms, how would I solve this $$\left.\frac12i\;\text{Log}\frac{1-i(1+e^{it})}{1+i(1+e^{it})}\right|_0^{2\pi}$$ would it equal; $$ \frac12i[ ln (\sqrt2) + I arg \frac{1-2i}{1+2i} ...
11
votes
2answers
111 views

How to prove $\large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$

I was given a challenge of sorting the following numbers. $\Large\sqrt[\pi]{e} < \sqrt[\pi]{\pi}<\sqrt[e]{e}< \sqrt[e]{\pi}$. After some work I was able to figure out the order. How can one ...
5
votes
4answers
84 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 ...
2
votes
2answers
38 views

How to compute $\lim_{x \rightarrow \infty}\dfrac{\ln(1+e^{\alpha x})}{\ln(1+e^{\beta x})}$? [closed]

I need to compute $\lim_{x \rightarrow \infty}\dfrac{\ln(1+e^{\alpha x})}{\ln(1+e^{\beta x})}$, where $α > 0$ and $β > 0$
0
votes
1answer
51 views

How to find the value of N in a logarithmic series

${\rm{log}}_2(1+p_1h_1)+{\rm{log}}_2(1+p_2h_2)+\cdots+{\rm{log}}_2(1+p_Nh_N)=NX$ Here $X$ and $h$ are known, but $N$ is an unknown value I want to find the value of $N$ Please note that ...
-2
votes
3answers
35 views

Logarithmic equation with negative component [closed]

What would be the solution for this since it has negative RHS? $$7 \times 7^{8v + 4.3} - 4 = 9$$
0
votes
0answers
14 views

$\{ \log f: f \in F \}$ is a compact of a space of holomorphic functions [closed]

Let $\log$ a principal logarithm branch. Let $U \subset \mathbb{C}$ a open subset and $F$ a compact subset of $H(U)$, where $H(U)$ is the space of holomorphic functions in $U$ with the topology $T$ ...
0
votes
2answers
22 views

Write the expression in terms of $\log x$ and $\log y$.

$$\log\Big(\frac{x^3}{10y}\Big)$$ Write the above expression in terms of $\log x$ and $\log y$. To be honest, I'm really unsure as to how the final answer should look like. In other words, ...
3
votes
2answers
48 views

How to solve the following equation? $\log_3\big(\log_x(\log_416)\big)=-1$.

$$\log_3\big(\log_x(\log_416)\big)=-1.$$ I am trying to solve this equation for $x$. This is what I have so far: $$\log_3(\log_x 2)= -1.$$ Okay, now I have this: log2 = (1/3)logx How do I isolate ...
3
votes
3answers
59 views

Logarithmic Equation: Solve for $x$

$$\log_{3x}81 = 2$$ How would I go about solving this? This is what I tried: $$\log_{3x}81 = 2$$ $$\frac{\log81}{\log 3 + \log x }= 2$$ Where do I go from here? If I isolate $\log x$ on one side, ...
2
votes
2answers
31 views

Solve the following exponential equation

$$7^{3x+1}=5^x$$ I am trying to solve this equation. I solved the equation and got what I believe to be the correct answer, but when I verify the answer it appears to be incorrect. Any idea why? Here ...
0
votes
1answer
34 views

Logarithm as a limit of a decreasing sequence

Let $k : \mathbb{N}^{>0} \to \mathbb{R}$ be such that $k_n = n(t^{1/n} - 1)$, where $t \in [0, 1] \subset \mathbb{R}$. Note that $\lim_{n \to \infty} k_n = \log(t)$. The plot of $k$ for a given $t$ ...
3
votes
0answers
15 views

limit of a sequence of iterated logarithms [duplicate]

I was playing around with the family of sequences $s(x)$ defined for $x > 0$ as $s(x)_0 = x$, $s(x)_{n+1} = \log(1+s(x)_n)$ and I noticed a strange behavior. Specifically, regardless of the choice ...
-2
votes
1answer
41 views

Is this identity correct?

Is this identity true? Wolfram|Alpha thinks is not. $$x^{ln(x^3)} = e^{3\,[ln(x)]^2}$$ That's how I demonstrated it: $${\left(e^{ln(x)}\right)}^{3\,ln(x)} = e^{3\,[ln(x)]^2}$$ ...
1
vote
2answers
27 views

Induction of factorial

I was perusing the wikipedia page on Mathematical induction, and it mentions it's possible to prove by induction that. $\frac{n^{n}}{3^{n}}<n!<\frac{n^{n}}{2^{n}}$ for $n\geq6$ Proof for $n=6$ ...
1
vote
1answer
22 views

Proof of discrete logarithm?

If you have that $a$ is a primitive root mod p. How can you prove this discrete logarithm property? $log_{a}(b_1b_2) = log_{a}(b_1) + log_{a}(b_2)$ (mod $p-1$) I see the proof for the regular ...
7
votes
3answers
96 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: $$ \frac12 \log(3+2 \sqrt2) = 0.881373587019543...$$ ...
2
votes
2answers
30 views

Calculate the following intergral

I have to calculate the following integral $$ \int \sqrt[3]{1+x\ln{x}} * (1+\ln{x}) dx$$ I have thought about using the following notation: $$ t = {1+x\ln(x)} => x\ln{x} = t-1 $$ But here I ...
1
vote
1answer
26 views

Simplify $log(sinhz)$ when $|z|$ tends to $0$?

I was given $\log(\sinh z)$ and I need to show it tends to $\log z$ when $|z|$ tends to $0$. I have tried converting $z$ to $x+iy$ then split $\sinh z$ but that doesn't seem to get me anywhere. I ...
0
votes
1answer
51 views

The limit of $ m\int_{a}^{1/m} \frac{dx}{x}=0 $ and $ m\int_{a}^{\infty} \frac{dx}{x^{1+m}}=0$ as $m\to0$

Given $ a >0 $ is it correct that $$ \lim_{m\to 0}m\int_{a}^{1/m} \frac{dx}{x}=0 $$ by the properties of the logarithm function? Or on the other hand, $$\lim_{m\to 0} m\int_{a}^{\infty} ...
0
votes
2answers
13 views

Adjusting the steepness of a curve

I've got an array of numbers, each between 1 and 0, sorted in descending order. When i put those numbers on a graph, they decrease too quickly: I would like to create a function ...
1
vote
1answer
22 views

prove logarithmic inequality for N>1200

For N > 1200 how can i prove that 3.09N/Log(N) + 1 <= 1.7(2N+1)/Log(2N+1) (sorry, could not figure out how to put the 'less than or equal' symbol there, tried \leq)
1
vote
3answers
42 views

Is this power rule true for the natural base?

Two questions 1) I was wondering if $e^{k \ln{x}}=k$ for any k. Is it? 2)To test I went to Maple and typed e^-ln(x) and it gave $e^{-ln(x)}$. I tried simplify and ...
3
votes
4answers
102 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!
2
votes
4answers
38 views

Logarithmic property justification

I saw this particular line slammed in a proof and it bothers me I can't understand why this is obvious and how would one justify this : $$ 7^{\log (n)} = n^{\log (7)} $$ Can anyone explain ?
0
votes
1answer
16 views

Pull log of a constant out of an integral

Can you pull the log of a constant out of an integral? Can the integral of ln(x/5)•dx become the integral of ln(x)•dx - ln(5) ?
1
vote
2answers
49 views

How to solve $(x-1)e^{-x} > 0.5$

As the title mentioned, how to solve $x$ from the equation: $$(x-1)e^{-x} > 0.5$$ How can I solve this analytically? This is a part of my homework and I got stuck to this equation. I'm also ...
5
votes
3answers
57 views

$\log_{10}(1+10^{-n})<10^{-n}$?

In a paper I was reading, this inequality: $$\log_{10}(1+10^{-n})<10^{-n}$$ came up with no explanation for why it's true. Does anyone have a proof for why this holds? Is there some basic logarithm ...
1
vote
4answers
25 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...
0
votes
0answers
23 views

log of summation - what to do when the magnitude of terms is unknown?

I know that ln(a+b) can be rewritten as ln(a) + ln(1+b/a) as long as a>c. Does this last requirement, which I don't really understand, prevent the use of such a tool in working out an equation that ...
14
votes
4answers
104 views

High School Advanced Functions: Clarifying log rules in a log equation - $\log(x^2) = 2$, Solve for x.

I got in an argument with my teacher for the possible solutions of x. From some sources i found that because x is squared, negative values should be possible; however, my teacher insists that: $$ ...
0
votes
1answer
52 views

What's the base of this logarithm?

I'm reading a scientific paper and an equation of the following form appears: x = y log (z). I know what y and z are in my own data set. How do I solve for x? I'm used to logarithms of the form ...
-1
votes
0answers
12 views

Solving For A Given Equation With Exponents

I'm unable to solve the following equation. The question asks: The population $p$ at time $t$ years is assumed to be: $p= {2800ae^{0.2t} \over 1+ae^{0.2t}}$ where $a$ is a constant. Given that $p ...