Questions related to real and complex logarithms.

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7
votes
4answers
32k views

How to figure out the log of a number without a calculator?

I have seen people look at log (several digit number) and rattle off the first couple of digits. I can get the value for small values (aka the popular or easy to know roots), but is there a formula. ...
1
vote
7answers
215 views

Is the equation $\log[A/(-B)]=\log[(-A)/B]$ valid?

A friend sent me these lines: $$\log[A / (-B)] = \log[(-A) / B]$$ $$\log(A) – \log(-B) = \log(-A) – \log(B)$$ $$\log(A) – [\log(-1) + \log(B)] = \log(-1) + \log(A) – \log(B)$$ $$\log(A) – \log(B) - ...
1
vote
3answers
159 views

Logarithmic non-integer fractional value

Would it be possible to show the breakdown of how $\log_4$ $32$ = $\frac{5}{2}?$ I have to come up w/ 11 more just like it & I'm not sure how you came up w/ the answer. Thank you!
8
votes
4answers
3k views

Inverse of the natural log function $y =\ln x$

Of course, it is a well known fact that the inverse of $y=\ln x$ (natural logarithm of x) is $e^x$. Assuming we haven't heard of the exponential function at all, how do we prove that the inverse of ...
3
votes
1answer
83 views

Does $\int \frac{1}{u} du = \ln|u| + C$ also work when $u$ is complex?

Does $\int \frac{1}{u} du = \ln|u| + C$ also work when $u$ is complex? I was taught this in calculus but I'm not sure if it generalizes to complex variables. Thank you!
4
votes
1answer
382 views

Super logarithmic inverse of tetration

What's the super logarithmic inverse of tetration for $\bf{^{2}{x}}$? Is it $slog^{x}_{2}$?
0
votes
0answers
65 views

Question about an asymptotic analysis proof in Ball Collision Decoding paper.

On page 21 of Daniel Bernstein's paper "Smaller decoding exponents: ball-collision decoding" he presents a proof that I have a few questions about. $P,Q,R,L$ and $W$ are all positive and close to ...
2
votes
1answer
119 views

Complex logarithm, my answer is wrong

I am trying to calculate $$\log(-1+i)$$ I have $$\log(-1+i) = \ln|(-1+i)| + i\operatorname{Arg}(-1+i)$$ $$ = \ln\sqrt2 + i3\pi/4$$ However when I checked that in matlab and wolfram alpha they have ...
1
vote
3answers
66 views

If $\log_{b}N$ is rational, what are the limitations on the possible values of $b$ and $N$?

If $\log_{b}N$ is rational, is there a set of values to which $b$ and $N$ must belong? Is there a set of values to which $b$ and $N$ cannot belong? Further, if it is presupposed that $b$ and $N$ are ...
5
votes
2answers
261 views

How can I solve $x^x = 5$ for $x$? [duplicate]

Possible Duplicate: Is $x^x=y$ solvable for $x$? I've been playing with this equation for a while now and can't figure out how to isolate $x$. I've gotten to $x \ln x = \ln 5$, which seems ...
3
votes
3answers
2k views

Simplify $n^{\log\log n / \log n}$

I am interested in solving logarithmic expressions but I cannot do this. what does this expression simplify to? $$n^{\log \log n/\log n}$$
0
votes
3answers
1k views

What are examples of legitimate usage of logarithmic scale when drawing a chart?

Quite often a chart is drawn using logarithmic scale for one axis (usually the the y-axis). This is often used for abuse when presenting information - logarithmic scale alters greatly how the values ...
1
vote
3answers
111 views

Formula to $\ln$ that holds on interval $x \geq 1$

In the Wikipedia we can find two formulas using power series to $\ln(x)$, but I would like a formula that holds on the interval $x \geq 1$ (or is possible to calculate $\ln(x)$ to $x \geq 1$ with the ...
1
vote
1answer
196 views

Numbering primes within a range.

$$n\ln n + n\ln\ln n−n < p_n < n\ln n+n\ln\ln n \mbox{ for } n\geq 6$$ This is the range where the $n$-th prime must lie. However sieving within this range generates a large number of primes. ...
2
votes
3answers
394 views

Square Root Of A Square Root Of A Square Root

Is there some way to determine how many times one must root a number and its subsequent roots until it is equal to the square root of two or of the root of a number less than two? sqrt(16)=4 ...
67
votes
16answers
10k views

How do you explain the concept of logarithm to a five year old?

Okay I understand that it cannot be explained to a 5 year old. But, how do you explain the logarithm to primary school students?
1
vote
1answer
1k views

Complex logarithm and derivatives

c. Show that $e^{\mathrm{Log}(z)}=z$ and use this to evaluate the derivative of the function $\mathrm{Log}(z)$. d. Is it true that $\log(e^z)=z$ for complex numbers $z$? Justify your answer. ...
2
votes
2answers
132 views

About the logarithm of the negative unit

$${e}^{iz} = \cos(z)+i\sin(z)$$ and $$e^{i\pi}=-1$$ But then $$\ln(-1)$$ can be infinite many numbers (positive and negative), as $z$ is the natural logarithm of that number and the solution to the ...
4
votes
1answer
392 views

summation of an infinite series involving arctan

I'm having problems with the following calculation. Let $a >0$ $$ \begin{align} & \sum_{n=1}^\infty \arctan \left(\frac{2a^2}{n^2}\right) = \text{Im} \sum_{n=1}^\infty \log \left( 1 + ...
2
votes
1answer
65 views

Logarithmic inconsistency when integrating

Consider following integral: $$13\int{\frac{1}{8x-4}dx}\tag{1}$$ By factorizing the denominator and then taking the factor outside the integral sign, it can be rewritten as ...
0
votes
1answer
92 views

How do I write a log function that intersects the axis at particular points?

I'm trying to write a log function that intersects both the x and y axis at 100. Through trial and error I have come up with this function, which is close to what I want. It appears to intersect the ...
3
votes
2answers
428 views

How to maximise this function: $p \log (1 + x) + q \log (1 − x)$?

I was trying a past paper from http://www.abacus.utwente.nl/tentamens/M%20-%20Stochastic%20Processes/1a%20-%20Stochastic%20Processes%20Februari%202007.pdf Hint: Use the fact that $p \log (1 + x) + q ...
1
vote
4answers
135 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
172 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 $H(x) = lg(\Theta(g(n)))$ $\Rightarrow H(x) = \Theta(lg(g(n)))$ ...
3
votes
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
4k 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
1k 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
165 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
221 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
2k 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
277 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
128 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
79 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 ...
4
votes
1answer
991 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
173 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
251 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
144 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
4k 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
505 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
847 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
199 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
161 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
167 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
554 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
152 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
5answers
284 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
92 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
529 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 ...