Questions related to real and complex logarithms.

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2
votes
2answers
63 views

Can I simplify $\log_3{n} \cdot 2^{\log_3{n}} \cdot n$

Is it possible to simplify $$\log_3{n} \cdot 2^{\log_3{n}} \cdot n$$ I am actually trying to find the Big-O notation for this equation. But if you don't know what it is, is it possible to simplify ...
0
votes
2answers
86 views

What is the calculus explanation for logarithmic function's odd chain rule behavior?

Doing some rudimentary mental differentiation in my fluid mechanics homework, I encountered the following derivative: $\dfrac{d}{dr}\ln(r/a)$ I applied the chain rule, said it equaled ...
3
votes
1answer
183 views

How to find the roots of $f(x)= \ln( \frac{x+1 }{x-2})$?

I can't solve this equation: $$\ln\left(\frac{x+1}{x-2}\right) = 0.$$ I do: $$\begin{align*} \ln \left( \frac{x+1}{x-2} \right)&=0\\ \frac{x+1}{x-2} &= 1 \\ x+1&=x-2 \\ ...
1
vote
1answer
150 views

Inequality involving $\log$

Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $$ whenever the left hand side is defined. If it helps, I ...
16
votes
3answers
1k views

Why isn't a harp in a logarithmic shape?

I was watching a harp, yesterday, and thought about the mathematics involved. I know that music is closely related to logarithms, because having a string or pipe twice as long produces the same note. ...
0
votes
2answers
122 views

Solving Logarithm, Can't Use Quadratic Formula

I have a logarithmic equation that I am meant to find the value of $x$ for: $2\log_{9}(x)$ = $1/2$ + $2\log_{9}(5x+18)$ I get as far as here when I realize I cannot use the quadratic formula. ...
6
votes
1answer
222 views

Complex Logs and Roots of Unity

I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...
2
votes
2answers
1k views

What does the Matlab loglog plots do that is useful?

If there is a exponential relationship $y = e^x$ and we take the logarithm of this we can see a linear relationship $\ln (y) = x$. So we could plot the logarithm of the y-axis values against the x ...
2
votes
4answers
264 views

How to evaluate $\lim\limits_{s\to\infty}\log s$?

I am stuck with applying limit at the following step, limit $$ \lim\limits_{s\to\infty}\log s. $$ Now I am unable to do anymore steps(I cant figure out how do I apply the limit and get a valid ...
2
votes
3answers
239 views

Plotting $\frac{1}{\ln x}$

I need assistance in plotting the graph of $\frac{1}{\ln x}$. wolframalpha gives this. How to plot this function (both real and imaginary part) using calculus?
0
votes
1answer
256 views

How to learn graph plots of math functions?

I really don't know how to we say that a log function would look like this or polynomial function would look like this. I know that if I have like $X + Y = c$, I can draw straight line by taking ...
3
votes
2answers
123 views

Where is the mistake in this logarithmic equation?

The problem is: Given this equation find the possible values of $x$. $$\log_{2}\left ( \frac{x+2}{x-1} \right )+\log_{2}\left [ (x+2)(x-1) \right ]=2$$ First I defined the domain of the ...
1
vote
1answer
107 views

Why is the following about logarithms true?

I was reading some algorithm's analysis and I came across the following in the proof: $\log_2(n+1) \le h \le 1 + \log_2(n) \implies h = \lceil \log_2(n+1)\rceil$ Here both $h$ and $n$ are integral. ...
1
vote
2answers
110 views

inverse function of $y=ax\ln(bx)$

let be the function $y=ax\log(bx)$, here $a$ and $b$ are constants and $\log$ is the natural logarithm how can i evaluate the inverse function of this in terms of the Lambert $W$-function ??
0
votes
2answers
76 views

Summation identity involving logarithm

I'm having trouble understanding why this identity holds: $$\sum_{k=0}^{(\log n) - 1} \frac{n}{\log (n - k)} + \theta(1) = \sum_{k=1}^{\log n} \frac{n}{k}+ \theta(1) $$ Any pointers to a proof ...
0
votes
1answer
78 views

What is the following Calculation about?

i'm going through some homework, but there is one thing i don't understand. Our task is to explain the following calculation: Given: $h(x) = \ln(x^4)$ I. $ \begin{align} h(x) = t(x) ...
6
votes
6answers
1k views

An alternative way to calculate $\log(x)$

How can I replace the $\log(x)$ function by simple math operators like $+,-,\div$, and $\times$? I am writing a computer code and I must use $\log(x)$ in it. However, the technology I am using does ...
4
votes
3answers
106 views

Why is $x^{\log_x n}=n$?

I'm currently doing a couple of exercises on logarithmic expressions, and I was a bit confused when presented with the following: $5^{\log_5 17}$. The answer is $17$, which is the argument of the ...
4
votes
1answer
592 views

Inverse of the polylogarithm

The polylogarithm can be defined using the power series $$ \operatorname{Li}_s(z) = \sum_{k=1}^\infty {z^k \over k^s}. $$ Contiguous polylogs have the ladder operators $$ \operatorname{Li}_{s+1}(z) ...
3
votes
3answers
182 views

Reducing an equation related to logarithms, $\pi$, and probability

In this question, I mentioned that, assuming the digits of pi are independently-random, then at some point in pi's expansion there will be a sequence of one million consecutive 0's. I decided to ...
2
votes
1answer
143 views

How is $\ln(\sin 2x)^{\ln x} = \ln x \cdot \sin 2x$?

In my textbook, there's something like: $$\ln(\sin 2x)^{\ln x} = \ln x \cdot \sin 2x$$ I thought it should be $$\ln x\cdot \ln(\sin 2x).$$
0
votes
2answers
384 views

How to scale data for a log graph

I'm writing a program to "printer plot" some data (printing "*" characters), and I'd like to scale the data on the vertical axis logarithmically (ie, semi-log chart). I have data values ranging from ...
2
votes
1answer
140 views

Solving for $y$ in $y = x \ln(y)$

I want to solve $y = x \ln(y)$ for $y$ in terms of $x$. Wolfram Alpha kindly produces this plot with the solution, $y = -x W(-\frac{1}{x})$, where $W$ is the Lambert function. However, that only ...
5
votes
1answer
466 views

Comparing Powers of Different Bases

How can I know if one power is bigger than the other when the bases are different? For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
6
votes
1answer
122 views

logarithms power equation

I got a home work question to solve the following: $$ 27x^2 < x^{\log_3x} $$ can any one please explain how to solve this type of equation? I have no idea what to do or what to search for.
8
votes
6answers
427 views

Has anyone talked themselves into understanding Euler's identity a bit?

What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
0
votes
2answers
115 views

How can I calculate limit of division of two logarithms

How can I calculate $\lim\limits_{n \to \infty} \frac{\log_{a} n}{\log_{b} n}$. Where $a$ and $b$ are two integers.
1
vote
0answers
70 views

lower bounds for maximum computing times for integer factorisation

Supposing that n were known to have two prime factors, and that the computer had a database of all the primes $<\sqrt{n}$. Then, unless n is square, one factor would be $<\sqrt{n}$. If an ...
0
votes
3answers
137 views

Very basic question about logarithm

I don't fully understand why you need to solve it this way... $$x^25\log(2x+1)+9(-5)\log(2x+1)=0 $$ $$(x^2-9)5\log(2x+1)=0$$
14
votes
3answers
550 views

challenging alternating infinite series involving $\ln$

I ran across an infinite series that is allegedly from a Chinese math contest. Evaluate: $\displaystyle\sum_{n=2}^{\infty}(-1)^{n}\ln\left(1-\frac{1}{n(n-1)}\right).$ I thought perhaps this ...
1
vote
1answer
149 views

Can you use a logarithm coefficient in a linear equation?

I have an equation that looks like $x+(\ln3)y+z=0$ where there's a natural logarithm as a coefficient. Is it possible to have this in a linear equation? I know that you cannot have a root or a product ...
1
vote
1answer
46 views

Find M, since $\log_5 M = 2\log_5 A - \log_5 B+2$

Find M, since $\log_5 M = 2\log_5 A - \log_5 B+2$ I tried this: The answer is in function of A and B. $\frac{\log_M M}{\log_M 5} = 2\frac{\log_M A}{\log_M 5} - \frac{\log_M B+2}{\log_M 5}$ ...
0
votes
1answer
70 views

Re-writing a logarithm to a power

Given: $$(4\ln x)^2$$ Is this simplified to $8\ln x$, (multiplying the expression by 2), $32\ln x$, (square $4$ ($16$), then $\ln x$ ($2\ln x$) and combine again), or something else? Just to be ...
4
votes
1answer
404 views

Help understanding this formula on mutual information (used in bioinformatics)

I'm a bit lost on understanding this formula in my bioinformatics text, and I appreciate any tips or advice. Mutual Information, $\operatorname{MI}(X; Y)$ is: $$ \mu = \sum_x \sum_y p(xy) ...
5
votes
3answers
252 views

Solving the equation $3^{5x-2}=8^{8x-9}$

I'm trying to solve the equation $$3^{5x-2}=8^{8x-9}.$$ I'm assuming I need to do some work with logarithms, but I don't know what to do. Thanks in advance!
3
votes
2answers
491 views

Spectral Centroid computation issue

I guess my problem is related to logarithmic <-> linear scales. I'm trying to create a colored wave form by using a Spectral centroid. So far I got the color but the scaling is incorrect. Why this ...
0
votes
2answers
90 views

Solving $7 = 8 - 2e^{-3k}$

So, this is an assignment my friend and I have for our homework: $7 = 8 - 2e^{-3k}$ And the solution should be: $\frac{1}{3\ln(2)}$ But, I have no idea how they got there. I tried doing: ...
1
vote
1answer
254 views

How do you prove the following inequality concerning complex Logarithms?

If $0<|w|<1/2$, then $2|w|/3<|\operatorname{Log}(1+w)|$ using power series and modulus inequalities.
2
votes
4answers
95 views

Show that these are equivalent $x-\ln|1+e^x| = -\ln|e^{-x}+1|$

Can anyone help me show that the following equations are equivalent? $$x-\ln|1+e^x| = -\ln|e^{-x}+1|$$ I'm having a little trouble. It should be an easy solution, where I take one equation, start ...
3
votes
1answer
165 views

Groups where discrete logarithm is hard

What are examples of groups, where DLP (discrete logarithm problem) is hard? Two obvious ones are: integers modulo $p$ ($p$ being prime) and elliptic curves over finite fields. What are the others?
1
vote
2answers
134 views

Logarithm question

Alright, I'm helping a friend, but can't seem to be able to crack this question : If $\log_3 20 = a$, $\log_3 15 = b$ then how do we represent with a,b $\log_2 360$?
2
votes
2answers
705 views

Difference between Logarithms of different bases

Every time i see a logarithmic function and if it so happens that i'am required to take the derivative or the integral of that particular function i get stumped and i tend to avoid that problem. What ...
1
vote
1answer
628 views

Extracting angular velocity tensor from orthogonal matrices

Let us suppose we have two orthogonal rotation matrices representing a three-dimensional rotations $$\mathbf{R}(t)$$ and $$\mathbf{R}(t+\Delta t)$$ How is it possible to extract the angular velocity ...
2
votes
3answers
234 views

How do I solve this equation involving a logarithm?

I'm running in circles and I don't understand how to do this. $$x\log(x) = 100$$ Where the $\log$ is in base $10$, I understand that $\log(y)=x$ is $10^x = y$. So is it the same for $x\log(x) = ...
5
votes
1answer
1k views

How does Knuth's algorithm for calculating logarithm work?

I had a look at Knuth's The Art of Computer Programming, book 1. In chapter 1, section 1.2.2, exercise 25, he presents the following algorithm for calculating logarithm: given $x\in[1,2)$, do the ...
12
votes
5answers
4k views

Calculate logarithms by hand

I'm thinking of making a table of logarithms ranging from 100-999 with 5 significant digits. By pen and paper that is. I'm doing this old school. What first came to mind was to use $\log(ab) = ...
25
votes
3answers
7k views

What algorithm is used by computers to calculate logarithms?

I would like to know how are logarithms calculated by computers. The GNU C library, for example, uses a call to the fyl2x() assembler instruction, which means that ...
5
votes
6answers
1k views

Is there any significance to the logarithm of a sum?

Many years ago, while working as a computer programmer, I tracked down a subtle bug in the software that we were using. Management had dispaired of finding the bug, but I pursued it in odd moments ...
-1
votes
2answers
820 views

Basic logarithm questions

What is the value of $5^{\log_{5} 2}$? If $m = y$, which expression is equivalent to $\log (100 m^2$)? Which base logarithm do I use? Is it self implied that its $\log_{10}$? How do I solve it? When ...
2
votes
2answers
95 views

Difference of Logarithms to form a quotient?

Write as a single logarithm: $\log_8(5) - 2\log_8(6)$ To my understanding; because they are the same base you can just evaluate $\log_8\left(\frac{\log(5)}{\log(6)}\right)$ which is shown on the ...