Questions related to real and complex logarithms.

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2answers
42 views

Simplifying logarithm question

Without worrying about the background, I have a question that asks to solve for n. Pardon my formatting, but it seems understandable this way for the time being until I edit it: $$4n^2 = 256 ...
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3answers
66 views

logarithmic and polynomial equation

I have the following $(1-a^x)/x=b$ Can this be solved for x ? (if yes, how, if not why) I have gotten to many forms, but can't seem to isolate x.
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1answer
38 views

Convergente of serie log. [closed]

I would ask some help to solve the next set and how to explain their sum $$ \sum \limits^{\infty }_{n=2}\frac{\log[(1\text{+}\frac{1}{n} )^{n}(n+1)]}{\log(n)^{n}\log\text{(}n+1)^{n+1}} ...
0
votes
0answers
7 views

negative sign in direction of wave propagation

Say I have a EM wave that goes in the Z direction and E=Eo*exp(-jkz). Why does the negative sign mean the wave travels in the +Z direction and exp(+jkz) means it travels in the -Z direction?
2
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2answers
45 views

The minimum value of $\log_{10}x+\log_x 10$

Notation: $\log:=\log_{10}$ $\log x+\log_x 10$ $=\log x+ \frac{1}{\log x}$ $=\log(x \cdot \frac{1}{x})$ $=\log 1$ $=0$ Is the process correct? I doubt this is wrong. Please help. ...
4
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2answers
246 views

If $x$ is rational, can $\log(1-x)/\log x$ be algebraic?

If $x$ is positive rational number less than $\frac{1}{2}$, can the following logarithmic expression be equivalent to an algebraic number, say $g$? $$\frac{\log(1-x)}{\log x} = g$$
2
votes
4answers
53 views

Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists the same as $\lim_{n \to \infty} (1+1/n)^n$ exists

My expanded question: Is showing $\lim_{z \to \infty} (1+\frac{1}{z})^z$ exists as $z$ goes through real values the same as $\lim_{n \to \infty} (1+\frac{1}{n})^n$ exists as $n$ goes through ...
1
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1answer
35 views

Difference between the formula of Roger Cotes and Euler

What was the difference between the formula that Roger cotes derived and that euler got? I mean to say that Euler got the following formula : $$e^{ix} = \cos x+i \sin x$$ And Cotes got the following ...
0
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2answers
34 views

A Simple Logarithm Question

Solve for $x$: $\log_2 (2x+8)=3$ Correct Solution: $2x+8=2^3$ $2x+8=8$ $2x=0$ $x=0$ Why doesn't this work: $\log_2 (2x+8)=3$ Expand: $\log_2(2x)+\log_28=3$ $\log_2(2x)+3=3$ $\log_2(2x)=0$ ...
7
votes
7answers
2k 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 ...
0
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3answers
61 views

How to find $\log{x}$ close to exact value in two digits with these methods?

I'm trying to find the result of $\log{x}$ (base 10) close to exact value in two digits with these methods: The methods below are doing by hand. I appreciate you all who already give answers for ...
0
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2answers
60 views
14
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2answers
286 views

Integral ${\large\int}_0^1\left(-\frac{\operatorname{li} x}x\right)^adx$

Let $\operatorname{li} x$ denote the logarithmic integral $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Consider the following parameterized integral: $$I(a)=\int_0^1\left(-\frac{\operatorname{li} ...
0
votes
1answer
49 views

What role does $1/\alpha$ play in the last integral?

If we examine the inverse function $f^{-1}=log_{10}$, the whole situation appears in a new light: $$\begin{align} log_{10}'(x)&=\frac1{f'\left(f^{-1}(x)\right)}\\ &=\frac1{\alpha\cdot ...
0
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1answer
38 views

Solve by separation of variables: $\frac{dx}{dy}y\ln|x| = \big(\frac{y+1}{x}\big)^2$

I need to solve the problem above using separation of variables. I got as far as the below but it seems too complex to be right. Am I wrong somewhere? Because I think my final answer needs to simplify ...
3
votes
3answers
83 views

Find sum of series [closed]

I need to find the sum of the following series: $$\sum_{n=2}^\infty \ln\left(1-\frac 1{n^2}\right)$$ How to proceed with this?
1
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1answer
515 views

logarithmic function between two points

I need to find the logarithmic curve between two points $$A(0,5),\quad B(180,9)$$ We know that the formula for logarithmic function is: $\;f(x) = \log(x)\,\;$so $$ 5 = \log(0),\quad 9 = \log(180)$$ ...
0
votes
2answers
31 views

Sequences identity

I have some problems to find a way to prove the following statement, if someone could give me any suggestions would be grateful: Show that $$ log\text{(}a_{n}+\text{1})\approx a_{n} $$ when $$ ...
1
vote
2answers
61 views

Evaluate $\log_{2005}(1/2)\log_{2004}(1/3)\log_{2003}(1/4)\ldots\log_2(1/2005)$ [closed]

The numbers 2005, 2004, 2003, ..., 2 are the bases. I cannot understand how to start the question. Please help. What to do in these type of questions? Thanks in advance.
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2answers
2k views

Graphing: Given two points on a graph, find the logarithmic function that passes through both.

Is there such a method to do this? I would like to come up with a logarithmic function (a graph that looks like a square root graph) that passes through two given points. Haven't had any luck in ...
1
vote
1answer
53 views

For which values of a parameter an equation has one Real root

The following equation is given $$\log_{x-1}(x^2+2ax) - \log_{x-1}(8x-6a-3)=0$$ And I am trying to find for which values of $a$ it has only one root, which is real. It is obvious that $$x-1>0 ...
0
votes
2answers
21 views

Logarithm where $0<a<\frac{1}{2}$. Find $x$

Given that $\log_a(3x-4a)+\log_a(3x)=\frac{2}{\log_2a}+\log_a(1-2a)$ where $0<a<\frac{1}{2}$. find the value of $x$. I got the attempt until $x=\frac{2(a+\sqrt{(a-1)^2}}{3}$ and ...
2
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1answer
49 views

Are these proofs of the 1st and 3rd Laws of Logarithms valid?

Disclaimer: I dont mean that I've discovered a conceptually completely different way of proving those laws, of course. I just found myself proving them like this and then realized that they're ...
5
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4answers
3k views

log base 1 of 1

What is $\log(1)$ to the base of $1$? My teacher says it is $1$. I beg to differ, I think it can be all real numbers! i.e., $1^x = 1$, where $x\in \mathbb{R}$. So I was wondering where I have gone ...
5
votes
4answers
457 views

Proof $e^x = \exp(x)$?

Define $$\ln (x) = \int^{x}_{1}\frac{1}{t}$$ Assume I have proven that $\ln x$ is one-to-one and therefore has an inverse $\exp (x)$. Define $e$ as: $\ln e = 1$ Now, if you have no other notion ...
7
votes
2answers
2k 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} ...
1
vote
0answers
35 views

How do I show that the integral $\int_0^\infty x^{-a} |\log x|^b dx$ only converges when $a = 1$ and $-2 < b < -1$?

This came up in a previous question, but was closed because the question wasn't terribly clear. I don't want to edit the other question substantially because it's not mine so I'm asking a new one and ...
2
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1answer
44 views

How to understand or derive the formula $Mantissa$ $bits$ $/$ ${log_2}$ $10$ = $Decimal$ $digits$ $of$ $precision$?

I asked a question a couple days ago about floating point precision on stackoverflow called, "Is floating point precision mutable or invariant?" and I received the following response. The formula ...
4
votes
5answers
54 views

Solving logarithmic equations including x

Let $$\log_3(x-2) = 6 - x$$ It's obvious drawing the graphs of the two functions that the only solution is $x=5$. But this is not really a proof, rather than observation. How do you prove it ...
3
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2answers
57 views

Regularizing the $\log\log n$ series

The divergent series $$\sum_{n=1}^\infty\log n$$ can be regularized using the derivative of the Riemann zeta function at $s=0$: ...
2
votes
3answers
42 views

Logs rules and Solving

I've got the equation : $$-1=\frac{-8e^{-t} + 3e^t}{2e^t}$$ I've moved some stuff around to get : $5e^t = 8e^{-t} $ But not sure where to go from here. Thanks for any help
2
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3answers
42 views

Set of real $a$ so that the inequality is defined but isn't true for a real $x$

$$x(x-\sqrt {4+\log_a7})\lt \log_7 \frac a{49}$$ I reach the interval $(0,1)$ after looking for the discriminant of the quadratic to be less than zero. However, the solution in the book is an ...
1
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1answer
18 views

Find distribution mean from the mean and sd of the log

I have a distribution with a long tail and use a model to predict the mean and standard deviation of its log. Given the mean and standard deviation of the log, how do I find the mean of the actual ...
1
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1answer
58 views

Derivative of $(\ln x)^e$ [duplicate]

In Randall Munroe's What If, he says that "if you want to be mean to first-year calculus students, you can ask them to take the derivative of $(lnx)^e$" He says, as I would expect, that the result ...
19
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4answers
4k views

“What if” math joke: the derivative of $\ln(x)^e$

Randall Munroe, the creator of xkcd in his latest book What if writes (p. 175) that the mathematical analog of the phrase "knock me over with a feather" is seeing the expression $ \ln( x )^{e}$. And ...
2
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1answer
31 views

Confused with integral and natural logarithm

When reading about ideal gas and adiabatic expansion, I got stuck with the following: $$W_{ab}=\int_{{\it V_a}}^{{\it V_b}}\!\,{\rm ...
17
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2answers
440 views

How can I find this limit involving thrice-iterated logarithm?

$$\lim_{x \to 0}\dfrac{\ln \ln \ln \left[x+(1+x)^{(1+x)^{1/x}/x}\right]+x\left[1-\dfrac{1}{e^{e+1}}\right]}{x^2}$$ How can I find the limit of this question? Any hint. Thank you so much.
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2answers
48 views

How to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$

I want to show that $\lim_{x \to 0} x^p (\ln x)^r = 0$ if $p > 1$ and $r \in \mathbb{N}$. To show this, I wanted to use that $\lim_{x \to 0} x \ln x = 0$, and in fact if $p \geq r$ we can write ...
0
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1answer
15 views

Integer solutions to the inequality $\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$

$$\log_{1/5}\log_3\frac {x-3}{x+3}\ge0$$ If $x$ is of the interval $[-8,10]$ Now I solved this, tried to limit $x$ as much as I could but I consistently get that there should be $10$ values of $x$ ...
2
votes
1answer
73 views

Solve for $x$: $x =\ln(x)^4$

I plotted the functions on both sides and it shows the equations has at least three solutions. Is there some non-interative (not sure if i used this term correctly - i mean the way you would solve, ...
3
votes
3answers
84 views

How is $O(\log(\log(n)))$ also $O( \log n)$?

How is $O(\log(\log(n)))$ also $O( \log n)$? I have seen this result somewhere with this but I still didn't quite understand how this is true. This would also help me compute Big Omega of the ...
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3answers
201 views

How to find the inverse of $n\log n$?

So I'm on chapter $1$ of introduction to algorithms & at the end the book proposes a problem: here The answers are there & I was able to work through most of them myself despite my lack of ...
0
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1answer
16 views

Question about a simple rule of the complex logarithm

According to the Wikipedia page on complex logarithms: Also, the identity $\log(xy) = \log x + \log y$ can fail: the two sides can differ by an integer multiple of $2\pi i$. Does the same hold ...
4
votes
1answer
47 views

Complex analysis integration with logs

$$\int_C \operatorname{Log}\left(1-\frac 1 z \right)\,dz$$ where $C$ is the circle $|z|=2$ I don't even know how you would begin doing this. I understand $\operatorname{Log}(z)=\ln|z|+i\arg(z)$, ...
1
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0answers
31 views

What properties does $a(n)$ have to fullfill to get $\log(2^{cn}+a(n))\sim\log(2^{cn})$?

Let $c$ be an exponential growth rate, and $a(n)$ any expression in $n$ (sequence, polynom, function,...). Consider $$ \log(2^{cn}+a(n)). $$ I am asking myself what properties (increasing, ...
0
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2answers
39 views

First order nonlinear ordinary differential equations

In my exercise I am stuck in a problem given below: $\ln\left(\frac{dy}{dx} \right) = x-y+1$ Although I could solve it if it was a linear equations. But ln() is a nightmare for me. Can anyone help me ...
2
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6answers
2k views

How do I square a logarithm?

How do I square $\log_2(3)$. Does it become $2\log_2(3)$ ? If yes,Please explain.
3
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3answers
90 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
0
votes
1answer
35 views

Converting an integrand into a polylog?

Compute the integral $$\int_0^1 dx\,dy\, \frac{\ln(1+y(1-x))}{1-xy}$$ I was just wondering if there is a way to convert the integrand into a polylog? This comes from a tutorial following a lecture ...
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2answers
26 views

Number of digits

I've been trying to find a solution to this problem for a while but I just can't seem to find the connection between the numbers and I really need help. I apologize if a problem like this one has ...