Questions related to real and complex logarithms.
5
votes
5answers
54 views
What is the limit of $\log_k(k^a + k^b)$ for $k \to +\infty$?
I'm not very good with analysis (I never studied it) but because of my "work" on other topics of mathematics I came to this problem.
$$\lim_{k \to +\infty }\log_k(k^a + k^b)=\max(a,b)$$
I'm sure ...
2
votes
3answers
89 views
A series converging (or not) to $\ln 2$
I have come across the following series, which I suspect converges to $\ln 2$:
$$\sum_{k=1}^\infty \frac{1}{4^k(2k)}\binom{2k}{k}.$$
I could not derive this series from some of the standard ...
3
votes
4answers
51 views
A general definition of Entropy (i.e. may or may not be expectation of the Log of the probabilities) [on hold]
Entropy may be defined as
Entropy = Σ G(p(x))
Where 'G' is any function that
goes asymptotically to plus infinity as it approaches zero from the positive side
and is monotonic between 0 and 1
...
0
votes
2answers
47 views
Proof the expession $\log_{12}{18}*log_{24}{54} + 5(\log_{12}{18}-log_{24}{54})=1$
I am trying to proof the following expression (without a calculator of course).
$\log_{12}{18}*\log_{24}{54} + 5(\log_{12}{18}-\log_{24}{54})=1$
I know this isn't a difficult task but it's just ...
1
vote
2answers
48 views
Definition of $a^b$ for complex numbers
Problem statement
Let $\Omega \subset C^*$ open and let $f:\Omega \to \mathbb C$ be a branch of logarithm, $b \in \mathbb C$, $a \in \Omega$. We define $a^b=e^{bf(a)}.$
$(i)$ Verify that if $b \in ...
2
votes
1answer
34 views
How to solve this logarithmic equation?
I want to solve this equation: $$8n^2 = 64n\log_{\ 2}(n)$$
After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to ...
2
votes
2answers
42 views
I need help on the process of solving this derivative.
How do I go about solving this derivative. $$f(x)=\ln\left(\frac{7x}{x+4}\right)$$
I go from this to $$1. \quad f(x)=\ln(7)+\ln(x)-\ln(x+4)$$ and then $$2. \quad f'(x)=\frac{1}{x}-\frac{1}{x+4}$$ then ...
2
votes
2answers
77 views
How do I simplify $\log (1/\sqrt{1000})$?
How do I simplify $\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right)$?
What I have done so far:
1) Used the difference property of logarithms
$$\log ...
0
votes
2answers
35 views
Help me solve this…
Assuming $a=\log 2$ and $b=\log 3$ (log is the base 10 logarithm). I have to find $\log_5 288$. How can I do this?
Edit: I've tried transforming $\log2$ to $\frac{\log_5 2}{\log_5 10}$ and same for ...
5
votes
3answers
141 views
Help with logarithmic definite integral: $\int_0^1\frac{1}{x}\ln{(x)}\ln^3{(1-x)}$
I'm look for a closed form evaluation of the following improper definite integral involving logarithms:
$$\begin{align}
I:&=\int_{0}^{1}\frac{1}{x}\ln{(x)}\ln^3{(1-x)}\,\mathrm{d}x\\
...
0
votes
0answers
22 views
Using math functions to time finales of a fireworks show
This year, I have the honor of programming two finales for a fireworks show.
I want to use math.
I suspect that I should use a function such as square root or log to specify the decreasing pause ...
0
votes
1answer
10 views
Consumption change calculation
I want to calculate yearly consumption change according to the following formula:
$$C_{t+1}=C_{t}e^{x_{t}}$$
I need to calculate ${x_{t}}$.
I have the consumption data $C_{t+1}$ and $C_{t}$.
-3
votes
2answers
45 views
How to get this answer [closed]
Anyone help me solve this question
$$\ln u + 2 \ln(1-u) - 2 \ln(1+u) = 2 \ln x + \ln c$$
I have the answer as $\frac{x y}{ (x^2 - y^2)^2} =c$, but I cant figure out how get this answer.
2
votes
1answer
29 views
Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?
Does the following have a closed formula?
$$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
7
votes
3answers
169 views
Is $ln(x)$ ever greater than $x$
Is $\forall x \in \mathbb{R}, \ln(x) \lt x$ a true statement?
Just wondering for some convergence related thing
2
votes
1answer
69 views
Checking derivation of y = a^x
Can you tell me if there are any flaws with this derivation of $y = a^x$...
The assumptions are that the derivative $$\frac{d}{dx}e^x = e^x$$ and that the derivative $$\frac{d}{dx}\ln x = ...
0
votes
1answer
39 views
Is there a property for log(n)/n?
I found a small exercise which I couldn't figure what to do, so I found a solution. Then I tried to understand it and everything went well until I got to this part:
$$\frac{1}{8} = ...
0
votes
2answers
46 views
Help me to solve math homework on logarithmic
How to solve this math home work? Please help..
What is the value of $\log \left(\dfrac{i\pi}{2}\right)$ ?
I got to know the answer is "$\dfrac{i\pi}{2}$", but don't know how to solve it. Please ...
1
vote
2answers
27 views
If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$
If $2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$, then find the numerical value of $\frac{x}{y}$
My try:
$2 \cdot log_e{(x -2y)} = log_e{y} + log_e{x}$
$log_e{(x-2y)^2} = log_e{xy}$
...
0
votes
1answer
24 views
Mathematics - geometric progression question
If $a$, $b$ and $c$ are in geometric progression, then what are $\log_ax$, $\log_bx$ and $\log_cx$ in?
What I did:
I substituted values for $x, a, b$ and $c$ and tried to solve it further.
What I ...
2
votes
1answer
55 views
What does this log notation mean?
Can someone please explain what $^2\log x$ means? Is it the same as saying $\log x^2$ or is it something completely different? Here is an image of it as an example:
0
votes
0answers
31 views
characteristic function of logarithm of random variable
If I know the characteristic function $\phi_X(t)$ of a random variable $X>0$, how can I write the characteristic function $\phi_Y(t)$ of $Y=\log(X)$?
I know that $\phi_X(t)=E[e^{itX}]$ and ...
5
votes
0answers
136 views
$\exp(\ln(x))=x$ and $\ln(\exp(y))=y$.
Let $(A,1_A,|\cdot{}|)$ be a unital Banach algebra, for instance $A=M_n(\Bbb R)$ or $M_n(\Bbb C)$. What is the union of all open unit balls $B_{\|\cdot{}\|}$ where $\|\cdot{}\|$ ranges over all ...
0
votes
0answers
16 views
Light intensity loss derivation [closed]
I was hoping somebody could help derive this known result:
Power change, $\gamma\ (\mbox{dB}) = -8.686a\ (\mbox{dB/km})$
from only:
power reduction in dB $= 10\log_{10}(\exp(-2aL))$
$a$ is given ...
0
votes
1answer
23 views
How to define this logarithmic function
I am trying to get my head around the definition of this function (that I concocted as an exercise in defining a function). Let $f$ denote the function satisfying:
$f(0) = +\infty$, and $f(+\infty) = ...
0
votes
1answer
45 views
Show that $\frac 1{\log_2x}+\frac 1{\log_3x}+\cdots+\frac 1{\log_{43}x}=\frac 1{\log_{43!}x}$ [closed]
Show that $\frac 1{\log_2x}+\frac 1{\log_3x}+\cdots+\frac 1{\log_{43}x}=\frac 1{\log_{43!}x}$.I am just not able to get it.please help.
0
votes
2answers
21 views
definition or property of logarithms
I've seen a lot of complicated logarithm definitions on this StackExchange and I have a rather simple question:
$$a^{b}=c \leftrightarrow \log_a{c}=b$$
Is this a definition of logarithms, which all ...
0
votes
2answers
34 views
Logorithms on a first level learning
Solve log$_{5x-1}$ $4$ $=$ $1/3$
$(5x-1)^{1/3}$=4
$((5x-1)^{1/3})^3$ = $4^3$
$5x-1=64$
$5x=65$
$13$
I am not sure where to go with this. I learned some things about logs before my class ended ...
0
votes
3answers
72 views
Question releating to the $\int^x_1\frac{\ln(t)}{t+1}$
If $f(x)=\int^x_1\frac{\ln(t)}{t+1}dt$ if $x > 0$. Compute $f(x) + f(1/x)$. As a check, you should obtain $f(2)+f(1/2)=(\ln2)^2$
I have tried evaluating the integral ...
2
votes
0answers
34 views
Interpolation of iterated logarithms
$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$
gives an interpolation between ...
3
votes
3answers
76 views
calculate $\int_{0}^{\pi} \int_{0}^{x}\log(\sin(x-y))dydx$
I was asked to find the integral $\iint_A \log(\sin(x-y))dxdy$ where $A$ is the triangle $y=0, x=\pi, y=x$ in the first quadrant.
I was given a hint: evaluate $\int_{0}^{\pi}\log(\sin(t))dt$ using ...
1
vote
1answer
44 views
$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$
I have to prove the following relations:
$\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a $
$\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $
I tried to use the relation that $\int_a^b \frac{1}{x} ...
0
votes
3answers
32 views
Logarithm deduction question
Given that $\log_{10}2 = 0.3010$ to four decimal places and that $10^{0.2} < 2$, is it possible to deduce that:
$2^{100}$ begins in a $1$ and is $30$ digits long;
$2^{100}$ begins in a $2$ and is ...
0
votes
1answer
32 views
How to calculate arithmetic mean of log values
I am working with really small values of probabilities and that is why their log values are used. So for example, let probA and probB be some normal values of probabilities of two events and because ...
1
vote
1answer
27 views
Help with Evaluating a Logarithm
A precalculus text asks us to evaluate $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}$
I do the following:
$\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{(2^2)^3\cdot 2^2}}{\sqrt[6]{2^3\cdot 2^2}}$
...
1
vote
3answers
49 views
Determine all positive numbers $a$ for which the curve $y = a^x$ intersects the line $y = x$ without calculus
The answer is $0 < a < e^{1/e}$ , but how to find it? Is it a system of equations? Which ones? I just need an idea at least, because I'm stuck.
If it is impossible without calculus, solve it ...
0
votes
2answers
38 views
Avoiding substraction for finite difference with log and exp
I want to approximate the derivative of f(x)
Finite difference
$f'(x) \approx \frac{f(x+h)-f(x)}{h}$
I was taught that the error from the substraction is blown up for small h. This I can verify ...
0
votes
1answer
14 views
How to interpret the difference in log points
How can we interpret the difference between two log points? Is it correct to interpret this difference in percentage points?
Thanks.
Marko
0
votes
1answer
29 views
Best way to handle the ratio which cannot be represented as floating point numbers.
I need to calculate the ratio of the form:
$s=\sum_1^3q_i$,$\quad$ $p_i=\frac{q_i}{\sum_1^3q_i}$,
where $q_i >0$. One problem is that $q_i$ are too small that they can not represented as ...
2
votes
4answers
446 views
Infinite sum of logarithms
Is there any closed form for this expression
$$
\sum_{n=0}^\infty\ln(n+x)
$$
1
vote
1answer
20 views
How can I read logarithmic scale?
I've got this histograms:
How can I read that logarithmic scale? For example, on the histogram 1 there is approximately $10^{-3}$ value at y-axis at 2 value at x-axis. Does it meant that there is a ...
1
vote
3answers
32 views
Find $log_{p}X^2$?
Given that $log_{p}X=5$ and $log_{p}Y=2$, find
i) $log_{p}X^2$
I did this, $X=p^5$ and $Y=p^2$
But how do I use them? Should I find $p$?
5
votes
3answers
59 views
Limit of logarithmic function using l'Hospital
How can I find the following limit:
$$\lim_{x\rightarrow \infty}\frac{\ln(1+\alpha x)}{\ln(\ln(1+\text{e}^{\beta x}))}$$
where $\alpha, \ \beta \in \mathbb{R}^+$.
My first guess was to use ...
0
votes
0answers
27 views
Do so big $p \in \Bbb N : \lim_{n \to \infty} \frac{\ln^p {n} }{n} = A \ne 0, A \in \Bbb R$ exist?
We know
$$\lim_{n \to \infty} \frac{\ln {n} }{n} = 0$$
$$\lim_{n \to \infty} \frac{\ln^n {n} }{n} = \lim_{n \to \infty} \frac{n\ln^{n-1} {n} }{1} = \infty$$
For usual $p \in \Bbb N $: $$\lim_{n \to ...
2
votes
0answers
96 views
What is ${\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$?
This is a new integral that I propose to evaluate in closed form:
$$ {\mathfrak{R}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$$
where $\log (z)$ denotes the principal value of ...
0
votes
1answer
21 views
How to clear variable $v$ from logarithmic equation
I have the following:
$6.4 = -\log\dfrac{5-v*0.1}{50+v}$
I would like to know how to solve the equation in order to get $v$'s value. Thank you very much.
2
votes
1answer
76 views
-ln(0.1) equalling to ln(10)?
I am having quite a headache wrapping my head around this solution. I do not understand the first line where they get lambda = ln(10) from statement to the left. Somebody please explain this to me. ...
0
votes
4answers
38 views
How do you solve this using only given values, logarithm rules and no calculator?
Given that $\log12=1.0792$ and $\log4=0.6021$, solve $\log8$ without a calculator.
I am familiar with the following three rules:
Product rule: $\log(a\cdot b)=\log a+\log b$
Quotient rule: ...
1
vote
0answers
17 views
Combining ±% with ±dB in measurement uncertainty
Firstly apologies if this is not the correct place to post this but wasn't sure which site would be good to ask regarding about measurement uncertainty calculation.
I am trying to calculate the ...
0
votes
2answers
68 views
What is the best way to calculate log without a calculator?
As the title states, I need to be able to calculate logs on paper without a calculator.
For example, how would I calculate $log(25)$
?
