Questions related to real and complex logarithms.

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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
186 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
156 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
45 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
31 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
52 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
133 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
69 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
40 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 ...
1
vote
1answer
26 views

Solving natural log equations

The equation $\ln(|y+1|)= x-2$ where you solve for $y$, I am just unsure of how the absolute value plays into this. I am assuming that I would convert to exponential form to get $|y+1|=e^{x-2}$ and ...
0
votes
1answer
35 views

LN word problem

measurement of a child's ability to learn is given by the function $$L(t)=\frac{ln(t+1)}{t+1}$$ where t is the child's age in years, for $0 ≤ t ≤ 5$ At what age does a child have the greatest ...
1
vote
2answers
48 views

( Logarithmic Equation ) Solve for x.

$(x+1)^{log(x+1)} = 100(x+1)$ Attempt at solution : $$ (x+1)^{log(x+1)} = 100(x+1)$$ $$= x^{log(x+1)} + 1 = 100x +1$$ $$=(x+1)+1=100x+1$$ $$=−98=99x$$ $$x=−98/99$$ But the answer given in the ...
1
vote
1answer
19 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
2
votes
2answers
596 views

How to recognise intuitively which functions grow faster asymptotically?

There are some cases where it is not so simple to decide which function grows faster asymptotically. For example, in the following cases, why (intuitively) $g(n)$ should grow faster than $f(n)$, or ...
0
votes
0answers
20 views

What would be the equation for “3% in 100 Hz range, 0.5% in the 2000 Hz range”

The smallest distinguishable pitch/frequency by a human ear is something like this: Pitch is our perceptual interpretation of frequency. As mentioned, ideal human hearing ranges from 20 to 20,000 ...
1
vote
2answers
66 views

Why does $\lim_{x\to0^{-}} \mathrm {Im}\left( \mathrm \ln \left(x\right)e^x\right)=\pi$?

Why does $$\lim_{x\to 0^{-}} \mathrm {Im} \left( \ln\left(x\right) e^x\right)=\pi$$ Obviously this is no coincidence. I was thinking maybe this has to do with Euler's formula, but I don't see how the ...
4
votes
2answers
66 views

Iterative calculation of $\log x$

Suppose one is given an initial approximation of $\log x$, $y_0$, so that: $$y_0 = \log x + \epsilon \approx \log x$$ Here, all that is known about $x$ is that $x>1$. Is there a general method of ...
3
votes
4answers
91 views

How to solve $\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$ without using L'Hospital's rule?

$$\lim_{x\to1}\left[\frac{x}{x-1}-\frac{1}{\ln(x)}\right]$$ How can I solve this without using the L'Hopital's rule? Any tips or hints would be greatly appreciated. I tried using the substitution ...
-1
votes
2answers
29 views

What is the Solution? x ≥8lgx

x ≥ 8lgx I have to find which x satisfy this inequality. I found the points using graph, but I'd like someone to show me how to find it without it.