Questions related to real and complex logarithms.

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0
votes
1answer
48 views

Finding value of (y) of logarithmic equation given (x)

I have an logarithmic equation $$\left[ r=a\,e^{b\,\theta} \right] $$ And I plot it to visualise it (see plot below). I can tell by the plot when (t=0), x1=0, y=1 (point AA) but how can I find out ...
0
votes
1answer
40 views

Expotential Growth/Decay - Problem Deriving Atmospheric Pressure Formula

I have a problem deriving the following formula: $$\frac{dP}{dh} = k\left(\frac{P}{T}\right)$$ Using the following 'rule': If $\ \dfrac{dA}{dt} = kA\,$ then $\,A = A_0\left(e^{\,kt}\right)\,$ ...
10
votes
1answer
76 views

Asymptotic behaviour of log log sum

I am interested in the asymptotics of $$F(m) := \prod_{j=1}^m \log(j+1) = \exp\left(\sum_{j=1}^m \log \log(j+1) \right)$$ Is there anything known? If not I figure I will need some good bounds on the ...
1
vote
2answers
78 views

Find the roots of a function with logarithms (possibly using lambert W function)

I am wondering if anyone can help me find an analytical solution to the roots of the following function: $$f(b) = c\log \left( \frac{b}{a} \right) + (n-c)\log \left( \frac{1-b}{1-a} \right),$$ $a,b ...
3
votes
1answer
108 views

Proof of Ramanujan's identity

I'm having trouble understanding Ramanujan's formula from his proof of Bertrand's postulate, namely: $$ \ln \lfloor x\rfloor!=\sum_{k=1}^{\infty}\psi\left(\frac{x}{k}\right) $$ where $ \ln x = ...
1
vote
2answers
58 views

how to solve this equation using logarithm, if not possible how to solve it?

how to solve the following equation: $$0.2948(1-(1+x)^{-5})=x$$ I know to satisfy this equation $x$ should be equal to 0.145 but how i can get there please help!
0
votes
3answers
57 views

Is $\log(-1)$ equal to $-\log(-1)$ [duplicate]

I thought it should be because if the logarithmic identities hold then, $$-\log(-1)=\log(-1^{-1})=\log(-1)$$ But $\log(-1)=i*\pi$ and $-\log(-1)=-i*\pi$
-2
votes
2answers
38 views

Can anyone simplify this logarithmic function? [closed]

The function is: $$X^{\log_a(Y)}$$ Thanks edit: This is from my book. The answer given is y* log (base a) X. But i cant solve. + I am from mobile and never used this site before sorry for ...
3
votes
0answers
32 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. ...
0
votes
1answer
41 views

Very strange logarithm simplification

I have very strange logarithm simplification: $$\begin{array}{l}\frac{{{{\log }_{{2^{{{(x + 1)}^2}}} - 1}}({{\log }_{2{x^2} + 2x + 3}}({x^2} - 2x))}}{{{{\log }_{{2^{{{(x + 1)}^2}}} - 1}}({x^2} + 6x + ...
2
votes
1answer
38 views

Find this sum $\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$ [closed]

Find $$\sum_{k=1}^{2015}k\lfloor \log_{2}{k}\rfloor\equiv \pmod {1000}$$ consider $2^{i-1}\le k\le 2^i-1$,then $\lfloor\log_{2}{k}\rfloor=i-1$,This problem have simple methods?
-3
votes
5answers
49 views

$ \log_{k}x \times \log_{5}k = \log_{x}5 $ find value of $x$. [closed]

If $ \log_{k}x \times \log_{5}k = \log_{x}5 $ ($k$ cannot be equal to $1$ and has to be more than $0$), then how do you find the value of $x$?
24
votes
6answers
3k views

The logarithm is non-linear! Or isn't it?

The logarithm is non-linear Almost unexceptionally, I hear people say that the logarithm was a non-linear function. If asked to prove this, they often do simething like this: $$\ln(x + y) \neq ...
4
votes
2answers
188 views

Integral $\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx$

How can we evaluate this definite integral $$I=\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx,$$ where $\displaystyle\phi=\frac{1+\sqrt5}2$ is the golden ratio?
3
votes
4answers
59 views

Using the $\ln(\cdot)$ for $(1-e^{-x})$

The given function: $$B= A(1-(e^{-x}))$$ Now, I want to 'destroy' the e-function by taking the logarithm of it. First, since $\ln(ab) = \ln(a) + \ln(b)$ we get that $\ln(b) = \ln(a) + ...
22
votes
2answers
2k views

Can a logarithm have a function as a base?

For example is $\log_{\sin(x)}(3x)$ a ridiculous equation? I couldn't find an example on any page about logarithms that used a function on a base, but it seems that for an equation like ...
4
votes
2answers
81 views

Ambiguity with Logs and Inequalities

Solve the following inequality: $$0.8^x > 0.4$$ Method 1 (Using the Common Logarithm) $$\log_{10}0.8^x > \log_{10}0.4$$ $$x\log_{10}0.8 > \log_{10}0.4$$ Because $$\log_{10}0.8 < 0$$ ...
0
votes
1answer
21 views

Predictability of what $\lfloor n\log n\rfloor$ “skips”

TL;DR: is there any way to tell what numbers will not be present in the function $\lfloor n\log n\rfloor$ under a given upperbound? I am writing a program that will calculate the sum of the gaps in ...
1
vote
4answers
48 views

How would I solve: $\log_{16} 32 = x$?

How would I solve: $\log_{16} 32 = x$? What I know: 16 is the base 32 is the exponent $$ 32 = 16^x $$ I'm stuck at this point$\ldots$
2
votes
2answers
57 views

Growth Rate $n\ln n$

I mistakenly posted this on MathOverflow. I hope this is a better place for it. I have been investigating a problem about sports teams and came across the function $n\ln(n)$. I want to see if I can ...
7
votes
1answer
157 views

Evaluating $\sum_{n \geq 1}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)$

Is there a direct way to evaluate the following series? $$ \sum_{n=1}^{\infty}\ln \!\left(1+\frac1{2n}\right) \!\ln\!\left(1+\frac1{2n+1}\right)=\frac12\ln^2 2. \tag1 $$ I've tried telescoping ...
1
vote
2answers
55 views

Inequality about logarithm

I have tried to prove the following inequality: $$ \left(1+\frac{\log n}{n}\right)^n \gt\frac{n+1}{2}, \mbox{for}\;n\in\{2,3,\ldots\} $$ which seems to be correct (confirmed by numerical result). ...
0
votes
0answers
40 views

How to divide a distance with tricky proportions

I have to divide a distance between two points (A, B), with specified proportions. Equations: $d_1 + d_2 = D$ $P_a * Log[\frac{4\pi d_1}{\lambda}] = P_b * Log[\frac{4\pi d_2}{\lambda}]$ $D, P_a, ...
-1
votes
2answers
46 views

Does limit exist for the following expression?

If limit exists, then what is its value? And if it does not exist then can we find where does this expression tends as $ n \to \infty$. The expression : $\lim\limits_{n \to \infty } ...
1
vote
1answer
49 views

Solving $s \le n\log n$ for smallest $n$

I am given an arbitrary positive integer $s$. I want to find the smallest integer $n$ such that $$s \leq n \log_2 n$$ where $\log_2$ is log base $2$. Is there an efficient way to compute $n$?
1
vote
1answer
75 views

Proof check $(\log\log n) /(\log n) $ approaches zero

Proof : If $|a| < 1$ then $(na^n)$ is a null sequence therefore if $b>1$ then ${n\over (b)^n}$ is a null sequence.There is always an $m$ such that for every $n > m$ $${n\over (b)^n} ...
0
votes
1answer
21 views

Will the absolute logarithm always produce the correct real result if one exists?

I'm a computer scientist, so my math skills are a bit rudimentary. The application I'm writing is more or less about solving equations. I'm only interested in real number solutions, so imaginary ...
4
votes
2answers
97 views

What is intresting about $\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}$?

Why does $$\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}=\frac{1}{\ln{x}}$$ There only seems to be a relation when using square roots, but not for cubed roots or ...
1
vote
0answers
34 views

(Numerical) Integration in log space

I have some function $f(x)$, which I'd like to integrate to find, $F(r) = \int_r^\infty f(x) dx $. Is there a way to do this using the values parametrized in log-space? I.e. some function $G(r, ...
3
votes
1answer
85 views

Alternative proof for the integral of $1/x$ being equal to $\ln (x)$?

This question relates to an attempt I made to answer the following question. I wonder if the approach below counts as an alternative way to show that $$\int_1^{1+x}\frac1t\ \mathsf dt = \ln(1+x).$$ I ...
2
votes
2answers
76 views

Root of Logarithmic Equation

I'm studying a function: $$f(x) = (x - 1) \log(x^2 - 1)$$ Having as first derivative: $$f'(x) = \frac{(x+1)\log(x^2-1) + 2x}{ x + 1 }$$ I'm looking for critical points ($f'(x) = 0$). I know it ...
1
vote
0answers
53 views

Linear to logarithm scale

I need to convert a linear range [0.1 ... 4 ... 10] to logarithmic one [0.5 ... 1 ... 5], so that 0.1 -> 0.5, 4 -> 1, 10 -> 5 I've found the similar problem here From half to double, linear to ...
1
vote
1answer
31 views

Rewriting approximated terms

The following data are inferred from a presentation slide, so I do not much info. Using linear approximation and log rules $\sqrt x $ can be rewritten as $\frac{x+1}{2}$, where $(1 \leq x \lt 2) $ ...
2
votes
1answer
45 views

Entropy of $f(x)=1$

Let $f(x)$ be a probability mass function $f(x) = 1$ on $x = [0,1]$, and entropy defined as $$H(p(x)) = -\int p(x) \log_2(p(x)) \, dx$$ where $p(x)$ is a pmf. Unless I've made an arithmetic error, the ...
3
votes
1answer
25 views

Solving equations with logarithmic exponent

I need to solve the equation : $\ln(x+2)+\ln(5)=\lg(2x+8)$ With the change of base formula we can turn this into: $\ln(x+2)+\ln(5)=\frac{\ln(2x+8)}{\ln(10)}$ We can also simplify the LHS with the ...
4
votes
7answers
135 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
2
votes
2answers
29 views

Positive entropy of system of mixed substance (mathematical viewopoint)

If two substances respectively having mass $m_1$ and $m_1$, constant pressure specific heat capacities $c_{p1}$ and $c_{p2}$, and temperatures $T_1$ and $T_1$ are mixed at constant pressure, reaching ...
2
votes
1answer
70 views

When is ${\log(a) \over \log(b)}$ an integer?

I've encountered this quite a bit. If I have ${\log(a)\over \log(b)} = c$ where $b$ is a known positive integer, what can be said about $a$ if $c$ needs to be an integer?
2
votes
5answers
135 views

Nice algebra question

The real numbers $x,y,z$ satisfy $$x+y+z=4$$ $$xy+yz+zx=2$$ $$xyz=1$$ Then $x^{3}+y^{3}+z^{3}$=??. It's for sharing a new ideas, thanks:)
-1
votes
1answer
57 views

Inequality with two variable [closed]

Proof this inequalities: $$\frac{x-y}{x} < \ln\frac{x}{y} < \frac{x-y}{y}, $$ if $0<x<y$. $$\frac{b-a}{\cos^2a} < \tan b-\tan a <\frac{b-a}{\cos^2b},$$ if $0<a<b<\pi/2$.
0
votes
3answers
45 views

Inequality $\left(\frac {17}{25}\right)^k \le 10^{-5}$ - Solve for $k$

How can I solve for $k$ the following inequality : $$ \left(\frac {17}{25}\right)^k \le 10^{-5} $$ This is what I got so far. By taking $\log_k$ from both sides I get: $$ \log_k{\left(\frac ...
0
votes
0answers
10 views

Pseudo-log transformation returning positive values

I'm looking for a transformation that acts similarly to natural log, but I want to return positive values only. My untransformed values range from 0.01 to 50. Of course I could simply offset the log ...
1
vote
1answer
51 views

For which value(s) of $b$, are $b\ln(x)$ and $b^{x} $tangent to each other?

For which positive value(s) of $b$, are $b\ln(x)$ and $b^{x} $tangent to each other? By setting the derivatives of the two curves equal to each other: $$\dfrac{b}{x}=b^{x}\cdot \ln(b)$$ ...
2
votes
4answers
67 views

Derivative of $e^{\sin(\ln[2 \arctan(2x)])}$ [closed]

Let $$F(x) = e^{\sin(\ln[2 \arctan(2x)])}$$ and take the first derivative of $F(x)$. Could someone walk me through each step here? It seems to be a pretty straightforward chain rule problem but it ...
2
votes
3answers
45 views

Minus sign in logarithm of integral's solution

I want to solve the following integral: $ \int \frac{dp}{a(1-p)u-bp} $ where $a$, $b$ and $u$ are some constants. After integration I get: $ p = -\frac{\log(-apu+au-bp)}{au+b} + C. $ According to ...
3
votes
2answers
31 views

Maximizing Theta in a Summation Formula

I need to take the first derivative of $$\sum Y_i (\log(\Theta )) +(n-\sum Y_i)(\log(1-\Theta )) $$ with respect to theta, and then solve for theta. I believe this is my derivative... $$\frac ...
2
votes
2answers
22 views

The number of logarithm applications to get from n below 1

Let $L(n)$ to be a number of logarithms that you need to apply on $n$ until you get below 1: $$ 0 \leq \log\cdots\log n < 1 \\ \uparrow \\ L(n)\mbox{-times} $$ Is there a name for this function? ...
1
vote
2answers
25 views

Branch points and natural maximal domain of $\log\frac{1+z}{1-z}$

So I have a function $\log\frac{1+z}{1-z}$ and I'm supposed to find its branch points, natural maximal domain. So far I converted $z$ to $x+iy$ but no real good. Can't check holomorphicity using ...
0
votes
1answer
42 views

Relation between number of digits of a number and its logarithm? [duplicate]

I found a couple of questions where, for example, they ask you to calculate the number of digits in $18^{200}$ and only the value of $\log 18$ is given. Can anyone tell me a way?
6
votes
3answers
106 views

Closed-form of $\int_0^1 x^n \operatorname{li}(x^m)\,dx$

I've conjectured, that for $n\geq0$ and $m\geq1$ integers $$ \int_0^1 x^n \operatorname{li}(x^m)\,dx \stackrel{?}{=} -\frac{1}{n+1}\ln\left(\frac{m+n+1}{m}\right), $$ where $\operatorname{li}$ is the ...