Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,059
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What property is being used to simplify this logarithm? $ 4^{log_2 n} = n^2$
I had this equation for calculating this recursion:
$$ T(n) = 4T(\frac{n}{2}) + n $$
I'm using a recursion tree to solve this, it's similar to this one:
But anyway what matters is that from the ...
- 33
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0
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13
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Applying log on a generating function
I am reading the Book SymmetricFunctions and Hall Polynomials by MacDonald and now reached the chapter "Power Sums" (Starting on Page 23). There we want to proof
$[P(t)=H'(t)/H(t)]$
In the ...
- 31
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78
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How do you solve this formula $(1 + 0.02X) ^{1/X} = 1.0161$
$X$ is supposed to be the duration of the loan in years.
The number $0.02$ stands for the nominal interest and the $1.0161$ stands for the compound interest + 1.
The equation can also be written : $(1 ...
- 131
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48
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Strange logarithm inequality
I was playing around with this function:
$$f(x)=\log_{\frac{1}{2}}(x^2-x-6)-\log_{\frac{1}{2}}\frac{x+2}{x-3}+4$$
I tried to find values of $x$ such that $f(x)\leq 0$. But for some reason, the answer ...
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1
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Find the value of $\log_{5}(0.0016)$
Find the value of $\log_{5}(0.0016)$
$y=\log_{5}(0.0016)=\log_{5}(0.2)^4 \Rightarrow 5^y=(0.2)^4$
$\Rightarrow \log5^y=\log(0.2)^4$
$\Rightarrow y\log5=4\log(\dfrac{1}{5})$
$\Rightarrow y\log5=4(\...
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54
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$\int\frac{\ln^kx\ln(1-x)}x dx$ Vs. $\int\frac{\ln x\ln^k(1-x)}x dx$
By using WolframAlpha, experimentely I observed that
$$\frac1{k!}\int\frac{\ln^kx\ln(1-x)}x dx=(-1)^{k+1}\text{Li}_{k+2}(x)+\sum_{i=1}^k \frac{(-1)^{k+1-i}}{i!} \text{Li}_{k+2-i}(x)\ln^{i}x+c.$$
On ...
- 8,524
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Computation of limit involving specific differences of logarithms
I wish to compute the following limit:
$$
\lim_{n\to\infty}\frac{1}{\ln(2)n}\left(\sum_{k=(2n/3)+1}^n \ln(k) - \sum_{k=1}^{n/3} \ln(k)\right).
$$
Putting the expression into a CAS just turns out the ...
2
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1
answer
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Iterating $\log(x\log(x\log(...)))$
For a real positive $x$, let $F(x)$ denote the sequence
$$
\left(\log x,\log(x\log(x)),\log(x\log(x\log(x))),\log(x\log(x\log(x\log(x)))),...\right),
$$
stopping at the first nonpositive value or ...
- 1,593
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Applications of the gamma function [closed]
I am a student studying logarithms and for a presentation about their uses I am demonstrating how the laws of logarithms can be used to extend the definition of x! to the reals. However, I am ...
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1
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When solving Logarithmic Inequalities why are the two following methods not the same?
When solving $\log_6 \left(\frac{12x-1}{x-3}\right)<0$, I would raise both sides as exponents to a base of $6$, resulting in $\frac{12x-1}{x-3}<1$, then move the $1$ to the left side, make a ...
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$x$ is distributed as Poisson with parameter $\lambda$, prove $\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0$
Let $x\sim\text{Pois}(\lambda)$, prove that for any $\lambda>0$,
$$f(\lambda)=\text{E}(\log(x+0.5)-\log\lambda)+0.02/\lambda>0.$$
In other words,
$$\sum_{x=0}^{\infty}\log(x+0.5)\frac{\lambda^x}{...
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Show that for $a \neq b$ it holds: $\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$ [duplicate]
Show that for $a \neq b$ it holds:
$$\frac{e^b-e^a}{b-a} < \frac{e^b+e^a}{2}$$
My first idea was to rearrange
$$2 \cdot (e^b-e^a) < (b-a)(e^b+e^a)$$
$$2e^b-2e^a < be^b + be^a - ae^b -e^a$$
...
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0
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Is there a better way to compute first digits of very large numbers?
The currently known method of finding the first digits of $a^b$ is multiplying $\log_{10} a$ by b, and extracting the fractional part. This allows us to compute the first digits of quite large numbers ...
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Manipulating Integrals involving Inverse exponential
I was studying integration when I came across mannipulating natural logarithms. It is known that:
$$
\int \frac{1}{x}dx = \ln{|x|} +c \forall x>0 \wedge \int \frac{1}{ax+b}dx =\frac{1}{a} {\ln{|ax+...
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2
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how to solve the following question using logarithms?
how can I use logarithms to compare a,b,c in an ascending order $$a=\frac{3^{8.7} - 3^{6.2}}{5}\\\
b=\frac{3^{11.7} - 3^{8.7}}{6} \\\
c=\frac{3^{11.7} - 3^{6.2}}{11} \\\ $$ i tried to simplify a b ...
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The polynomial P(x) is equal to $(i + x)^{2024}$. Let $S$ be the sum of all the real coefficients of $P(x)$. Find $\log_{2} S$. [closed]
I think the answer to this problem is $1012$, but I'm not sure.
Any ideas? Any and all help is appreciated.
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Quick doubt on this function domain
$$f(x) = \frac{e^{1-\ln(x-x^2)}}{\ln(1 - e^{x-x^2})}$$
I was solving this, and I found the domain is the emptyset.
Yet, then I checked with Mathematica and it returned me a different domain. When I ...
- 2,696
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23
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How do I find the logarithm and exponential of a string of powers of x (including non-integer powers) in terms of x?
I am trying to find a way to separate the negative and positive powers of x in the solution to the exponential and logarithm of real powers of x, where $M_L,M_E\subset \Bbb{R}$:
$$y=\ln\left(\sum_{k \...
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2
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1
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85
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What is $\int \frac{-2}{x^2-1}\,dx$
For $x\in\mathbb{R}$, what is the value of
$$
\int \frac{-2}{x^2-1}\,dx?
$$
Using partial fractions, we get
$$
\int \frac{-2}{x^2-1}\,dx= \int \frac{1}{x+1}\,dx+ \int \frac{-1}{x-1}\,dx=\log|x+1|-\...
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Differentiate this function: L1=ln(c11 +δλ11)+β1ln(c12 +δλ12)+Zλ11(1−n11)−αn11 [closed]
Differentiate this function: L1=ln(c11 +δλ11)+β1ln(c12 +δλ12)+Zλ11(1−n11)−αn11
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Implication in Logarithmic Inequalities
I am trying to understand an implication I came across in a research article, and I was wondering if someone could provide more details or clarifications on the reasoning behind it.
The inequality in ...
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How to approximate the median of the numbers in the first $n$ rows of Pascal's triangle?
How can we approximate the median of the numbers in the first $n$ rows of Pascal's triangle? (The top row is the $0$th row.)
Using Excel, I made a graph of the natural log of the median against $n$, ...
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Solving $\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$ [closed]
Here's the question I came across, they're inverses in this case, but I imagine that there is a way to do that without them being inverses.
$$\ln\left(\frac{1}{x-2}\right)=\frac{1+2e^x}{e^x}$$
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Logarithm inequality less than 1
I'm currently working on proving the inequality
$$
\frac{\log(\log(m))}{\log(\log(n))} \left(1 + \frac{\log(n/m)}{\log(n) \log(\log(m))}\right) < 1
$$
where $n > m$, and I suspect that using the ...
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Show that $ \lim_{x \to -\infty} (1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n ) = 0$
Let $x$ be real and define the entire function $f(x)$ as
$$ f(x) = 1 + \sum_{n=1}^{\infty} (\frac{x}{\ln^2(n+1)})^n $$
Now we have that
$$ \lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} (1 + \sum_{...
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If $m(A)=0$, does this imply that $m(\log(A)) = 0$ when $A\subset \mathbb{R}_{+}$
The question is as follows:
Let $A\subset \mathbb{R}_{+}$ and let $\log(A)=\{\log(t): t\in A\}$.If $m(A)=0$,then is it true that $m(\log(A)) = 0$? If $m(A) < \infty$, then is it true that $m(A) <...
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Why is $\ln2\approx 0.4^{0.4}$?
Just stumbled upon
$$\ln2\approx 0.4^{0.4}$$
and wondered if that's just a coincidence, or whether there's some deeper reason?
$$\ln2 - 0.4^{0.4}\approx 0.00000234$$
which is a relative error of just $...
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Simplifying $a\cdot\ln(x) + b\cdot\ln(y)$ as $\ln(x^a\cdot y^b)$
I ran into a problem where i need to simplify the equation $p\cdot\ln(x) + (1-p)\ln(y)$.
Can I solve it by moving the multiplier on top of the log as such: $\ln(x)^p + \ln(y)^{(1-p)}$ and then using ...
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On bounding complex logarithm times complex power function
In my textbook (Ordinary Differential Equations by Andersson and Böiers), it is claimed that the complex modulus of $(\log(z))^jz^\lambda$, where $j\geq0$ is a natural number and $\lambda$ a complex ...
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Solving $\frac{8^x-2^x}{6^x-3^x} = 2$ [duplicate]
$\frac{8^x-2^x}{6^x-3^x} = 2$
I was able to use difference of cubes on the numerator, and setting $a= 2^x$ would give the following:
$\frac{a(a-1)(a+1)}{3^x(a-1)} = 2$
Now we have a common factor and ...
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How to integrate $\int \ln(x+\ln(x+\ln(x+...))) dx$?
How to integrate $$\int \ln(x+\ln(x+\ln(x+...))) dx$$ ? I am trying to integrate this expression by taking the above expression as $y$. Therefore, $y=\ln(x+\ln(x+\ln(x+...)))$. Now from this we can ...
- 1,132
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How to log a distribution?
My understanding: Equation $(5)$ can be obtained when we let $e^x=\left({e^t}/{\theta}\right)^{\beta}$, which $x=\beta\,(t-\log\theta)$ , and substitute into $(4)$.
My question: Why equation $(7)$ is $...
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I am facing problem while solving the integrals $\int e^{\ln(x)}dx$ and $\int e^{\ln(|x|)}dx$.
I am facing problem while solving the integrals
$$I=\int e^{\ln(x)}dx$$ and
$$I_1=\int e^{\ln(|x|)}dx$$
In the case of $I_1$, $|x|$ will come in the place of $e$ and $e$ will go in the place of $|x|$. ...
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How to solve $10x=3^x+3$
How to solve $10x=3^x+3$?
There should be two answers from the graph plotted, one should be 3, but how do I get the other one?
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$\int \frac{-1}{x-1}\,dx$ vs $-\int \frac{1}{x-1}\,dx$
Consider the two integrals
$$
\int \frac{-1}{x-1}\,dx\text{ and }-\int \frac{1}{x-1}\,dx
$$
I would expect the solution to be $-\log|x-1|+C$, but I might be missing a detail here. Mathematica gives, ...
- 3,095
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Is $e^{\ln(-7) }= -7$?
I was going over some homework, and stumbled across a true or false question that presented as the following:
$e^{\ln(-7)}$ = -7, True or False
Seeing as the definition of a logarithm presents the ...
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Solve these simultaneous equations: logx + logy = log2, and x^2 + y^2 = 5 [closed]
How could I solve these equations simultaneously?
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Let $D$ be a domain, $a,b \in \mathbb{C}\setminus D$ and $f(z)=\frac{z-a}{z-b}$. Then there exists a holomorphic branch of the log of $f$ on $D$.
Let $D\subset \mathbb{C}$ be a domain and define $f: D \rightarrow \mathbb{C}$ by $f(z)=\frac{z-a}{z-b}$, where $a,b \in \mathbb{C}\setminus D$ are in the same connected component of $\mathbb{C}\...
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120
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Prove $\frac{1}{2}\ln(1+n)<\sin(\frac{1}{2})+\sin(\frac{1}{4})+\sin(\frac{1}{6})+ ...+\sin(\frac{1}{2n})<\frac{1}{2}\ln(n)+\ln(2)?$
Problem
Prove that $$\frac{1}{2}\ln(1+n) < \sin\left(\frac{1}{2}\right) + \sin\left(\frac{1}{4}\right) + \sin\left(\frac{1}{6}\right) + ... + \sin\left(\frac{1}{2n}\right) < \frac{1}{2}\ln(n) + \...
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0
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0
answers
25
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Calculating the Derivative of a Complex Expression [duplicate]
I am trying to find the derivative of the following expression:
$y = 1 + \frac{a}{1/x - a} + \frac{b/x}{(1/x - a)(1/x - b)} + \frac{c/x^2}{(1/x - a)(1/x - b)(1/x - c)}$
I have attempted to use basic ...
1
vote
1
answer
63
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Solving Logarithmic Expression
In the context of the thermodynamics of mixing two separate gases at the same temperature and pressure, one has the generic equation,
\begin{gather*}
-\frac{\Delta S_{mix}}{nR} = X_A \ln(X_A) + X_B \...
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votes
3
answers
263
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Prove that $2 \int_{x}^{x+1} \log(t) dt \geq \log(x (x+1))$.
Prove that $2 \int_{x}^{x+1} \log(t) \,dt \geq \log(x (x+1))$.
I want proof of this without using geometry. I need to know the techniques to solve this other than geometry.
One can easily see this ...
2
votes
1
answer
94
views
Solving $\log_3x^3 - 4\log_9x - 5\log_{27}x^{1/2} = \log_94$ [closed]
Having trouble solving this simple logarithm problem:
$$\log_3(x^3) - 4\log_9(x) - 5\log_{27}(x^{1/2}) = \log_9(4)$$
I’ve been stuck as I when I solve it I get an answer of $2$, when the actual ...
0
votes
4
answers
127
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How to determine the almost accurate initial guess to calculate the $\ln(x)$ via Newton-Raphson method?
I'm developing my own Real Number class, and for that, I have to implement the Natural Logarithm function.
I've used Maclaurin series of Natural log with iteration of 300 or 500. The division method ...
0
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0
answers
42
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Compute a tight upper bound of $\sum_{i=1}^{n-1}\frac{1}{3^i\log{n}- 3i}$?
I am trying to compute a tight upper bound of the sum below.
$\sum_{i=1}^{n-1}n\frac{\frac{1}{3^i}}{\log_3{(n/3^i)}}$
I was able to 'simplify' it up to the expression below.
$n\sum_{i=1}^{n-1}\frac{1}{...
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3
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$\frac{\Gamma\left(x+\frac{1}{x}+1\right)}{2}\geq \left(\Gamma(x+1)\Gamma\left(\frac{1}{x}+1\right)\right)x^{-\frac{1}{2\pi}\ln\left(x\right)},x>0$
It's a very challenging and accurate inequality let's say :
Let $x>0$ then we have :
$$\frac{\Gamma\left(x+\frac{1}{x}+1\right)}{2}\geq \left(\Gamma(x+1)\Gamma\left(\frac{1}{x}+1\right)\right)x^{-\...
- 3,456
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1
answer
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Showing $ (2/3)^{\log_{3/2}(n/n_0)} =n_0/n $
I read the following identity in the CLRS Introduction to Algorithms book, and I can't work out the computation.
$$
(2/3)^{\log_{3/2}(n/n_0)} =n_0/n
$$
I tried to expand the exponent using the ...
- 185
-1
votes
1
answer
90
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Calculate $i^{\log(−𝑖)}$ (principal value) [closed]
I am working on a complex number problem and need some guidance on how to calculate the principal value of the expression "$i^{\log(-i)}$". I understand that the principal value of a complex ...
1
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1
answer
135
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Extraordinary Numbers
Can you please explain what are Extraordinary Numbers in detail? At the same time, I would also like to confirm whether the equivalent problem of Riemann Hypothesis mentioned here is correct (like it'...
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