Questions related to real and complex logarithms.

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2
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1answer
60 views

Growth rate of Taylor convergents near pole

For any fixed $z_0\in\mathbb{C}\setminus \{0\}$ and $\beta\in\mathbb{R}^{+}$, prove that $$\left.T_n\left(\log^{\beta}z;z_0\right)\right|_{z=0}\sim\log^{\beta} n$$ Note: I observed that this holds ...
3
votes
1answer
499 views

contour integration of logarithm function

I'm new to contour integral involving branch point and stuck on this particular integration. Here is the problem: $$\int_{\mathcal{C}}\log z\,\mathrm{d}z,$$ where $\mathcal{C}$ is a closed square ...
1
vote
1answer
151 views

Attempted exercise using Littlewood's theorem

This was an exercise to try to show we can use Littlewood's theorem$^1$ to prove that $$\lim_{N \to \infty}\frac{1}{N}\sum_{n=1}^{N} \frac{g(n)}{\log p_n} = 1 \hspace{30mm}(1)$$ If $\vartheta(p_k) ...
1
vote
0answers
96 views

Expectation of functions of binomial random variable involving logarithms

Let $X\sim\text{Binomial}(n,p)$ where $n$ is the number of trials and $p$ the probability of success of each trial. I am trying to evaluate the expected value of the following functions of $X$: ...
0
votes
1answer
75 views

A problem of Logarithm

Find the minimum value of $$\frac{\log_bc}{\log a} + \frac{\log_ca}{\log b} + \frac{\log_ab}{\log c}$$ since I do not know how to write log base a index a so I gave it in that manner. I tried out the ...
0
votes
3answers
161 views

Proof of logarithm power change

I am not too sure how to explain this in words. So the question is proofing that $a ^{\log_bc} = c ^{\log_b a}$ So far what I have done was: I cannot think of anything else, I mean if I do the ...
1
vote
1answer
663 views

Indices, surds and logarithms equation

Can we use indices or using logarithms is better ? $$\dfrac{5^x-5^{-x}}{2} = 3$$ Solve $x$ correct to $4$ decimal places.
0
votes
1answer
58 views

What is meant by “show the following relations over $A$” in this problem?

Will someone explain what is being asked for by "show the following relations over $A$"? Let $A = \{1, 2, 3, 4, 5, 6\}$ Show the following relations over $A$. 1. $R1 = \{(x, y) \mid \log2 x < ...
3
votes
4answers
121 views

Solving for $x$: $3^x + 3^{x+2} = 5^{2x-1}$

$3^x + 3^{x+2} = 5^{2x-1}$ Pretty lost on this one. I tried to take the natural log of both sides but did not get the result that I desire. I have the answer but I would like to be pointed in the ...
1
vote
1answer
73 views

Why is this limit $\frac{e^x}{x^{x-1}}$coming out wrong?

Attempting to answer this question, I thought to evaluate the limit by taking the logarithm and then using L'Hopital's rule: $$\begin{align} L&=\lim_{x\to\infty}\dfrac{e^x}{x^{x-1}}\\ ...
1
vote
1answer
97 views

How to numerically integrate expression using its log transform

I am trying to use numerical integration to compute the integral $\int_0^1 f(\rho) d\rho$ where $f(\rho)$ is calculated as $\ln f(\rho)$ for improved numerical precision and stability. At the moment ...
0
votes
1answer
66 views

How can I easily compute the $\log_{60/47}3$?

I'm working on a recurrence problem which needs me to simplify $\log_{60/47}3$? Since $\frac{60}{47}\approx 1.2765$, how would I know how much of that I need to multiply by itself to get $3$? I have ...
4
votes
4answers
122 views

Simple looking log problem

How would I solve this for $x$? The original problem is $$x+x^{\log_{2}3}=x^{\log_{2}5}$$ I have tried to reduce it down to this, $$x^{\log_{10}3}+x^{\log_{10}2}=x^{\log_{10}5}$$ I have been ...
0
votes
1answer
537 views

Integral of $z^{n} \log z $ on the unit circle under two assumptions

I'm asked to calculate $\int_{|z| = 1} z^{n} \log z dz$ in two ways: (1) if $\log 1 = 0$; (2) if $\log (-1) = i \pi$. I understand it means that in case (1) I have to work with the principal ...
1
vote
2answers
68 views

Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number?

Given $\{\log_ab \mid a,b\in \mathbb N, \mathrm{gcd}(a,b)=1,a,b≥3\}$ does the sum of any two $\log_ab$ form an irrational or rational number? I know that $\log_ab$ is irrational, but does the sum of ...
3
votes
1answer
113 views

Show that map is conformal

I want to show that the map $\phi(r,\theta) = r^\lambda (\cos(\lambda \theta), \sin(\lambda \theta))$, where $\lambda \in \mathbb{C}$, is conformal on the slit plane $\{(r,\theta)| r > 0, -\pi < ...
1
vote
1answer
67 views

Solving logarithmic equation

I'm having trouble solving this equation. I know there is a solution as my graphics calculator can solve it, but I want to see the steps on how to get the answer. The mathematical equation is: ...
0
votes
2answers
64 views

determine x in $x\log_\frac{1}{10}(x^2+x+1)>0$

I wanted to know, how can i determine the values of x for which $x\log_\frac{1}{10}(x^2+x+1)>0$ going to the question, we must have $x>0$ and $\log_\frac{1}{10}(x^2+x+1)>0$ or both must ...
1
vote
1answer
200 views

Littlewood's 1914 proof relating to Skewes' number

From Littlewood's 1914 theorem (paraphrase): I propose to show there are arbitrarily large values of x for which successively $\psi(x) - x < - K\sqrt{x}\log\log\log x \tag{A}$ $ ...
4
votes
1answer
813 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
0
votes
1answer
133 views

Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
0
votes
1answer
677 views

Finding original amount in half-life problem

Say the half-life of an element is 1590 years. If 10g of the element is left after 1000 years, how much was there originally?
1
vote
1answer
58 views

How can I calculate this exponential growth?

I'm reading the book "Singularity is near", and there is a passage where the author says: "It takes 100 years to achieve this, with current rate of progress, but because we're doubling the rate of ...
-1
votes
1answer
150 views

Doubling Time for certain bacteria

Say a culture of bacteria doubles in weight every 24 hours. If it originally weighed 10g, what would be its weight after 18 hours? I know how to calculate half-life but don't know about doubling ...
0
votes
1answer
31 views

Rounding to the nearest term in a geometric progression

Consider the following progression: where i is ith number within the progression. I would like to devise an equation that will round input value to the nearest number from this progression. For ...
1
vote
2answers
74 views

Solving equation with logarithms

I happen to use this heavy math for the first time for a long time (if ever) and don't know how to bite it. Given: $$\begin{align} A &= 1.45\\ B &= 4.1\\ C &= 14\\ ...
0
votes
1answer
68 views

Calculating half-life?

After taking a tablet, a patient has 10 units/ml in a sample of blood taken soon after, and this decreased to 6 units/ml 9 hours later. What is the half-life of the tablet? How long will it take ...
2
votes
2answers
184 views

Confused with natural logarithms

How can we solve the following natural logarithms? I'm confused with this stuff: $\ln(x+1) - \ln x = \ln 3$ $\ln(x+1) + \ln x = \ln 2$
3
votes
3answers
153 views

Solving a simple exponential equation

How can I solve this logarithm: $$ e^{2x} - 3 e^x + 2 = 0. $$ I think it should be re-written as a quadratic equation in $e^x$.
2
votes
4answers
202 views

What log law justifies $(\lg n)^{\lg n} = n^{\lg \lg n}$?

I was reading the solution to 3.2-4 on this blog (cropped image pasted here) notice the person says $\frac{(\lg n)^{\lg n}}{n} = \frac{n^{\lg \lg n}}{n}$ What log law justifies that? Also, is it ...
3
votes
2answers
432 views

Prove that if $a^x = b^y = (ab)^{xy}$, then $x + y = 1$

The question is prove that if $a^x = b^y = (ab)^{xy}$, then $$x + y = 1$$ I've tried: $$a^x = (ab)^{xy}$$ $$\log_aa^x = \log_a(ab)^{xy}$$ $$x = xy \log_ab $$ $$y^{-1} = \log_ab$$ but then I get ...
1
vote
2answers
166 views

Solve for x, when $ \log_3 (2 - 3x) = \log_9 (6x^2 - 19x + 2)$

How do you deal with the different bases when solving the equation: $$\log_3 (2 - 3x) = \log_9 (6x^2 - 19x + 2)$$ I'm going round in circles trying to reconcile the bases.
7
votes
2answers
163 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 ...
0
votes
1answer
153 views

Johann Bernoulli did not fully understand logarithms?

This wikipedia article makes the claim: "Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms." This is found under ...
0
votes
2answers
123 views

How to go from a sum to a product and a product to a sum?

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
2
votes
3answers
99 views

Proof that $ \lim_{x \to \infty} x \cdot \log(\frac{x+1}{x+10})$ is $-9$

Given this limit: $$ \lim_{x \to \infty} x \cdot \log\left(\frac{x+1}{x+10}\right) $$ I may use this trick: $$ \frac{x+1}{x+1} = \frac{x+1}{x} \cdot \frac{x}{x+10} $$ So I will have: $$ ...
3
votes
1answer
113 views

log transformation for dummies

I have a question which is probaly very simple to answer for most people here: We have a formula: y = -log(x) Then this happens to x: ...
23
votes
2answers
514 views

Last $n$ digits of $a^b$

Last year I came acoss the following problem in a mathematics competition What are the last $2$ digits of $2012^{2012}$? $\ \ \ \ \ $(Ans: 56) I found the last two digits using the standard ...
4
votes
2answers
150 views

Limiting value of $\lim \frac{1}{k}\sum_{n=1}^k \frac{p(n+1)-p(n)}{\log p(n)}$

Empirically it seems $$\lim_{k\to \infty} \frac{1}{k}\sum_{n=1}^k \frac{g(n)}{\log p(n)} = 1\tag{1} $$ in which p(n) is the nth prime and g(n) is the prime gap $p(n+1)-p(n).$ Cramer conjectured ...
3
votes
2answers
58 views

Logarithm inequality for vectors

I am trying to prove the following result. Let $d$ be a vector in $\mathbf{R}^{n}$ with $\|d\|_{\infty} < 1$. Then, $$ \sum_{i=1}^{n} \log(1 + d_{i}) \geq \mathbf{1}^{T} d - \frac{\|d\|_{2}^{2}}{2 ...
3
votes
0answers
89 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b ...
1
vote
1answer
79 views

The bound of a log function

It looks like we can control $\log\frac{1+z}{1-z}$ by $\log\frac{1+r}{1-r}$ if $|z|=r<1$ where the logarithm is defined on the branch obtained by deleting the negative imaginary axis. I tried to ...
2
votes
1answer
65 views

Evaluating: $\int^{n}_{1}[\ln(x) - \ln(\lfloor x \rfloor)] dx $

I am attempting to evaluate the integral: $$\int^{n}_{1}\ln(x) - \ln(\lfloor x \rfloor) dx $$ To a form: $$f(x) + O(g(x))$$ where $g(x) \rightarrow 0$ as $x \rightarrow \infty $ How do I compute ...
5
votes
3answers
224 views

Computing the value of logarithmic series: $Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $

Given a series of the type: $$Q(s,n) = \ln(1)^s + \ln(2)^s + \ln(3)^s + \cdots+ \ln(n)^s $$ How does one evaluate it? Something I noticed was: $$Q(1,n) = \ln(1) + \ln(2) + \ln(3)+ \cdots+\ln(n) = ...
0
votes
1answer
97 views

Question about natural logarithm in the exponent of the e-function

I wonder which rule dictates that e^(-2x+ln(c)) is equal to e^(-2x) * c I know that the logarithm naturalis is the "reverse-function" of the e-function but why isn't it e^(-2x) + c instead?
3
votes
2answers
760 views

Stuck on an 'advanced logarithm problem': $2 \log_2 x - \log_2 (x - \tfrac1 2) = \log_3 3$

I'm stuck on solving what my textbook calls an "advanced logarithm problem". Basically, it's a logarithmic equation with logarithms of different bases on either side. My exercise looks like this: $$2 ...
2
votes
1answer
48 views

Generalised logaritmic function

I was wondering if there was a function that extends the domain of the following function to non-negative real numbers. For non-negative integer $n$ and real $y$, $y = f(x,n)$ is given by: $$f(x,n) = ...
1
vote
1answer
160 views

How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$

Solve the following equation for $x$ : $ \quad3^{2x+1}-10\cdot 3^x+3=0 $ I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how ...
3
votes
2answers
1k views

How to solve equations with logarithms, like this: $ ax + b\log(x) + c=0$

I encountered an equation of type $$ ax + b\log(x) + c=0$$ Here a, b, and c are constants. Does anyone know how to solve these type of equations? I guess this way: $$\log(x)= \frac{c-ax}{b}$$ $$x= ...
0
votes
3answers
2k views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$