Questions related to real and complex logarithms.

learn more… | top users | synonyms

1
vote
4answers
134 views

$\log_2$ approximation in $[1,2)$

this is realistically for a programming project, but is more math centric then CS centric. I am attempting to write a function that approximates a power function, but in order to complete I need to ...
0
votes
1answer
168 views

$\Theta$-notation of a logarithm

Given $H(x) = lg(f(n))$, where $f(n)$ is an asymptotically positive function, is it always true that if $f(n) = \Theta(g(n))$, then $H(x) = lg(\Theta(g(n)))$ $\Rightarrow H(x) = \Theta(lg(g(n)))$ ...
1
vote
2answers
3k views

How do I determine the increasing order of growth of a set of functions?

I've been struggling with a homework-like problem (I'm a self-studier) and having trouble understanding the 'hints' I've already received. Please forgive my lack of knowledge around terminology. ...
0
votes
1answer
3k views

Casio fx-82MS: n-th logarithm

I have got a Casio fx-82MS calculator, and I want to calcuate various logarithms with it, especially $\log₂(x)$-style logarithms. Unfortunately, its user interface contains buttons for just $\ln$ and ...
-3
votes
2answers
99 views

How do I know $\frac{\ln{n}}{n}\ge\frac{1}{n}$?

How do I know $$\frac{\ln{n}}{n}\ge\frac{1}{n}$$
6
votes
1answer
894 views

How many digits does $2^{1000}$ contain?

I tried this way, I only need to know if this is correct or if there are better ways to solve this: $2^{1000}$ does not have a factor of $5$ obviously therefore we can assume $$ 10^{m} < 2^{1000} ...
2
votes
1answer
163 views

Wondering if anyone knows how to prove this $y =(\log 2)^{y}$

A value of $y=.5295431$ does satisfy the equation $$ y = (\log 2)^{y}$$ But I havn't seen any ways to prove it. $\log$ is base $10$ and $\ln$ is $\log$ to the base $e$ Note: I would like to see a ...
2
votes
0answers
209 views

Exponential average of logs

I'm working on a sound recognition algorithm where an "exponential moving average" is used for "adapting" to sound levels. It turns out that taking an average of logs works better than simple sums ...
1
vote
1answer
1k views

Lambert W / Product log function?

I would like to solve this equation: $$n \cdot 2^n = 15000$$ And according to WolframAlpha $$n=\frac{W(15000\log(2))}{\log(2)}, \text{ where }\log\text{ is ln}$$ Which shows that I need to use the ...
0
votes
1answer
226 views

How To Create a Non-Linear Output from a Linear Input?

I'm not even sure how to ask this question, so bear with me for a second. Given a linear input value, such as floating point numbers between 0 and 1, how can I produce an output that favors higher ...
0
votes
1answer
124 views

Is $\lim_{x\to \infty} \frac{\ln{x}}{x} =\lim_{x\to \infty} \frac{\frac{1}{x}}{1}$?

In my notes its given $$\lim_{x\to \infty} \frac{\ln{x}}{x} = \lim_{x\to \infty} \frac{\frac{1}{x}}{1}$$ Is that correct? How do I get that? I think another example is also related ...
2
votes
2answers
78 views

How can I apply log laws here?

Solve for $x$: $$ 2^{2x+1} - (17)2^x + 8 = 0 $$ I have the answers: -1, 3 I tried a few different transformations, but couldn't get a clear answer. I suspect that I am overlooking a property of ...
3
votes
1answer
797 views

Comparing Powers with Different Bases Using Logarithms?

I looked all over to see if a question like this had already been answered, but I couldn't find it. So here goes: I need a general formula for comparing two (insanely huge) powers. I'm pretty sure ...
2
votes
1answer
158 views

How to determine periodicity of complex log in different bases?

How do you determine the "period" of a complex logarithm as a multivalued function in an arbitrary (real or complex) base? I apologize in advance if my terminology is incorrect, but let me illustrate ...
1
vote
2answers
231 views

Steps to calculate $\log_2\, 0.667$

This could be a basic question. But I would like to know steps I should follow to calculate $\log_2\, 0.667$. EDIT In an answer I found it says $(0.038 \log_2 0.038) = -0.181$. How this calculation ...
2
votes
2answers
2k views

Why are logarithms not defined for 0 and negatives?

I can raise $0$ to the power of one, and I would get $0$. Also $-1$ to the power of $3$ would give me $-1$. I think only some logarithms (e.g log to the base $10$) aren't defined for $0$ and ...
1
vote
1answer
139 views

what is the application of log(x) where x is negative number

what is the application of log(x) where x is negative number? Anyone knows real usecase?
1
vote
2answers
3k views

Why does Wolfram Alpha handle $\log$ and $\ln$ the same?

I thought $\log(n)$ was like $100^x = n$ and $\ln(n)$ was $e^x = n$. But when I do $\ln(80)$, it gives me the answer for $\log$. Why is that?
1
vote
1answer
320 views

simple calculation using logs

Suppose we are comparing implementations of insertion sort and merge sort on the same machine. For inputs of size $n\in\mathbb{N}$, insertion sort runs in $8n^2$ steps, while merge sort runs in ...
2
votes
3answers
792 views

Logarithms explained simply

Sorry for the trivial question. If I have the expression $\log(5)$, and the base is $10$, what operation is being performed on the number $5$, in words? For example, I know that exponents work (say ...
0
votes
2answers
198 views

How do you solve this simple logarithm problem?

I'm comparing efficiencies for the famous fake-coin algorithms. Specifically, I'm looking at a two-pile approach and a three-pile approach for a solution. I have found that, like a binary search, ...
6
votes
3answers
160 views

Evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$.

How would one evaluate $\int_0^1 {\ln(1+x)\over x}\,dx$? I'd like to do this without approximations. Not quite sure where to start. What really bothers me is that I came across this while reviewing ...
1
vote
1answer
163 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...
2
votes
3answers
429 views

How can I solve $8n^2 = 64n\,\log_2(n)$

I currently try to analyze the runtime behaviour of several algorithms. However, I want to know for which integral values $n$ the first algorithm is better ($f(n)$ is smaller) and for which the second ...
2
votes
1answer
147 views

Solving logarithmic equations of of the form $\ln(xa)= b\ln(c-x)$

Given: $$\ln(xa)= b\ln(c-x)$$ I am unsure of how to manipulate the values within the natural logs to solve for x while the factor b remains. I can safely move in circles by applying the definition ...
4
votes
4answers
264 views

How to find the limit $\lim \limits_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right]$

Solve $\space \begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] \end{align*}$. I did this way: $$\begin{align*} \lim_ {x \to+\infty} \left [ \frac{4 \ln(x+1)}{x}\right] ...
0
votes
1answer
85 views

Equation model for project effort.

It is my first time here, so I hope I'm keeping on topic. I wanted to find an equation where I could use the variables which affect the amount of work, in a way that feeling it the variables, I'd ...
2
votes
6answers
487 views

Proof that $\int_1^x \frac{1}{t} dt$ is $\ln(x)$

A logarithm of base b for x is defined as the number u such that $b^u=x$. Thus, the logarithm with base $e$ gives us a $u$ such that $e^u=b$. In the presentations that I have come across, the author ...
1
vote
2answers
521 views

Why are $\log$ and $\ln$ being used interchangeably?

A definition for complex logarithm that I am looking at in a book is as follows - $\log z = \ln r + i(\theta + 2n\pi)$ Why is it $\log z = \ldots$ and not $\ln z = \ldots$? Surely the base of the ...
6
votes
4answers
239 views

Solving the exponential equation: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$

I have this exponential equation that I don't know how to solve: $3 \cdot 2^{2x+2} - 35 \cdot 6^x + 2 \cdot 9^{x+1} = 0$ with $x \in \mathbb{R}$ I tried to factor out a term, but it does not help. ...
14
votes
3answers
452 views

Are the logarithms in number theory natural?

I find the frequent emergence of logarithms and even nested logarithms in number theory, especially the prime number counting business, somewhat unsettling. What is the reason for them? Has it maybe ...
0
votes
1answer
49 views

Formula to calculate when you will have a certain amount of money in your bank account

What is the formula to calculate when I will have a million dollars in my bank account? An example is that I have $\$6,000$ in my account and have $6\%$ interest rate on that. How long will it take ...
4
votes
2answers
255 views

Can all logarithm problems be solved algebraically?

Trying to solve $\log_2(x-1)=\log_3(x+1)$ and can't seem to get it algebraically. Tried changing bases, moving things around, but can't seem to crack it.
4
votes
1answer
308 views

The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
1
vote
3answers
10k views

From natural log to log base 10

The constraints of this question is related to a programming problem, but I must get the math right in order for it to be applied to code. The actual problem is I need a function that evaluates to log ...
5
votes
2answers
439 views

continuum between linear and logarithmic

A friend and I are working on a heatmap representing some population numbers. Initially we used a linear color scale by default. Then, because the numbers covered a wide range, I tried using a log ...
6
votes
2answers
203 views

Prove $ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2 $

Prove that $$ n+1<\frac{\log 4}{\log3}+\frac{\log 44}{\log33}+\frac{\log4444}{\log3333}+\frac{\log 44444444}{\log33333333}+\cdots+\frac{\log 444\ldots444}{\log333\ldots333} <n+2$$ where last ...
4
votes
1answer
92 views

Does the logistic function really uniquely satisfy this?

It is said that the logistic function (denoted $y(u)$ below) is derived from the relation: $$\frac{dy}{du}=y(u)(1-y(u))$$ Does $y(u)=\frac{1}{1+e^{-u}}$ really uniquely satisfy this? I don't see ...
2
votes
1answer
431 views

closed form solution for summation of $\log(i)$

Is there a way to find a closed form solution for: (Note that base is $2$) $\displaystyle\sum_{i=1}^n\log_2(i)$ thanks for any help Can't find a formula for this
1
vote
2answers
268 views

Graph of a Log Function

I am curious as to why Wolfram|Alpha is graphing a logarithm the way that it is. I was always taught that a graph of a basic logarithm function $\log{x}$ should look like this: However, ...
0
votes
1answer
304 views

Reverse of a log-based f(x)?

I am attempting to create a function, f(y) that can reverse my existing function, f(x). ...
2
votes
1answer
160 views

Are all logarithms multiple of each other?

I was doing a time complexity problem, and the solution mentioned that there is a single class for logs. Ie. we can write $\log_a (x) = \Theta(\log_b(x))$ where $a$ is not equal to $b$. This can be ...
5
votes
1answer
107 views

Solutions for this logarithmic equation.

For which values of $k$ does the equation $\log_a(kx+3)+\log_a(x+1)=\log_a(2x+1)$ have one or more solutions in $x$? The logarithmic functions must have the restriction that the argument is ...
2
votes
3answers
323 views

Logarithms of the form $x=e^y$

I have the following math problem: The number of people in a town of 10,000 who have heard a rumor started by a small group of people is given by the following function: $N(t) = ...
1
vote
3answers
1k views

Solve 10 base logarithms

I'm a n00b in math and I wanted to know how should I solve ten base logarithms. e. g.: Log 40 to base 10 Thanks in advance.
2
votes
3answers
206 views

Solving $\int\frac{\ln(1+e^x)}{e^x} \space dx$

I'm trying to solve this integral. $$\int\frac{\ln(1+e^x)}{e^x} \space dx$$ I try to solve it using partial integration twice, but then I get to this point (where $t = e^x$ and $dx = \frac{1}{t} ...
1
vote
3answers
112 views

Comparing numbers in form $x^y$

Let's consider two numbers in form $x_1^{y_1}$ and $x_2^{y_2}$ How can we compare those two numbers without evaluating them ? Can we use logarithms to check it ? If yes - how ? Thanks in advance. ...
5
votes
3answers
273 views

Does $\log(ab)^n$ equal $(\log(a)+\log(b))^n$ or $n\log(a)+n\log(b)$?

I think this might be a case of slight ambiguity in notation, but here goes: On a test question, I was required to expand the expression $\log (ab)^n$. Since the logarithm is a function, I reasoned ...
1
vote
3answers
946 views

arithmetic progression involving logarithm

$\log_2 X$, $\log_2 (X+9)$ and $\log_2(X+45)$ are 3 consecutive terms of an arithmetic progression; find $\qquad$(i) the value of X; $\qquad$(ii) the first term and the common difference; and ...
3
votes
2answers
106 views

Limit with prime sequence and inverse logintegral

I found formula below$$\lim_{n\to\infty}\frac{\operatorname{li^{-1}}(n)}{p_n}=1$$ where $\operatorname{li^{-1}}(n)$ is inverse logintegral function and $p_n$ is prime number sequence. Can anyone ...