Questions related to real and complex logarithms.

learn more… | top users | synonyms

0
votes
1answer
467 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
111 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
64 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
172 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
707 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
128 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
572 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
55 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
123 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
30 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
65 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
167 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
389 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
164 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
156 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
105 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
118 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
98 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
101 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
489 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
148 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
78 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
63 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
210 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
95 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
601 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
154 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
974 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
1k 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? $2\leq x <\infty$
4
votes
2answers
145 views

Estimating $\sum_{p_2 \leq x} (\log p_2)^2$

This was an exercise to use the approach here to estimate the sum $\sum_{p_2 \leq x} \log (p_2)^2,$ in which $p_2$ are numbers containing two prime factors (repetitions allowed). $\pi_2(x)$ is the ...
1
vote
2answers
99 views

Filling in 'x' in a log function

if $3^5=x$ (exponential equation) converts to log form gives $log_3x=5$ that makes sense. $$ 3^5 = 243 \Rightarrow x=243 $$ So if I take the log form again: $log_3x=5$ and replace $x$ with $243$. I ...
3
votes
4answers
2k views

Is it possible to solve logarithm when base is unknown?

Is it possible to solve logarithm equation when the base of the logarithm is unknown but the result is known. Here is an example: $$ \log_{X} (\frac{223}{150}) = 20 $$ This basically means that if x ...
0
votes
3answers
517 views

Ln Series Summation

I have been given: $$\ln{n}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\text{ for sufficiently large }n$$ Which I can equate to $\ln(n)=\sum\limits_{i=1}^n \frac{1}{i}$ The series I need ...
2
votes
3answers
69 views

Evaluating $\lim\limits_{x\to 0^{+}} \frac{x}{\ln^2 x}$

How can I find: $$\lim_{x\to 0^+} \frac{x}{\ln^2 x} $$ I know that the limit is $0$. I tried sandwich theorem but I don't know what could be bigger. Thanks in advance.
4
votes
3answers
215 views

Can we *ever* use certain log/exp identities in the complex case?

This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
1
vote
2answers
1k views

Finding Big-O with Fractions

I'd want to know how I can find the lowest integer n such that f(x) is big-O($x^n$) for a) $f(x) = \frac {x^4 + x^2 + 1}{x^3 + 1}$ I've fooled around with this a bit and tried going from $\frac ...
22
votes
1answer
446 views

The positive root of the transcendental equation $\ln x-\sqrt{x-1}+1=0$

I numerically solved the transcendental equation $$\ln x-\sqrt{x-1}+1=0$$ and obtained an approximate value of its positive real root $$x \approx 14.498719188878466465738532142574796767250306535...$$ ...
4
votes
2answers
96 views

What is an effective and practical means to teach about natural logarithms and log laws to high school students?

My students are quite practically minded, and I have found that teaching them concepts in a practical manner to be very helpful (maths 'experiments'; modelling on the smartboard etc). I am looking ...
1
vote
3answers
119 views

What does ($\ln x$) or ($\log x$) mean?

How does a logarithm followed by a variable read such as ($\ln x$) or ($\log x$). Is it $\log$ times $x$ or the $\log$ of $x$? I'm a little confused by this...?
0
votes
1answer
41 views

What role does 1/α play in the last integal?

Link of the page:http://imgur.com/N4uUeA9 α is defined as follows:http://imgur.com/6ESJUE8 Why is it here?I can't make any sense out of its use.
9
votes
4answers
812 views

Prove that $\log _5 7 < \sqrt 2.$

Prove that $\log _5 7 < \sqrt 2.$ Trial : Here $\log _5 7 < \sqrt 2 \implies 5^\sqrt 2 <7.$ But I don't know how to prove this. Please help.
2
votes
6answers
125 views

Help with differentiation of natural logarithm

Find $\;\dfrac{dy}{dx}\;$ given $y=\frac{\ln(8x)}{8x}$. The answer is $\;\dfrac{1-\ln(8x)}{8x^2}\;$. Can you show the process of how this is worked? Thanks.