Questions tagged [logarithms]
Questions related to real and complex logarithms.
10,256
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how to evaluate this integral $\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$
Question statement: how to evaluate this integral $$\int_0^1 \frac{\ln x \, \text{Li}_2(1-x)}{2+x} \, dx$$
I don't know if there is a closed form for this integral or not.
Here is my attempt to solve ...
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2
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Logarithmic inequality involving $a_1, a_2, ..., a_n$
Given the real numbers $a_1, a_2,...,a_n$ all greater than $1$, such that $\prod_{i=1}^{n} a_i=10^n$, prove that:
$$\frac{\log_{10}a_1}{(1+\log_{10}a_1)^2}+\frac{\log_{10}a_2}{(1+\log_{10}a_1 + \log_{...
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How to show $\log(z) = \log(r) + i \theta$ without implicitly assuming $z = r \exp (i \theta)$ - from Penrose Road to Reality
In Roger Penrose's book Road to Reality - Chapter 5 - he goes to great lengths to arrive at the standard polar expression for a complex number $w = r e^{i \theta}$ via a discussion of complex ...
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Reducing product of powers of logarithm
I am trying to show that
$$(\log(a))^n (\log(b))^m = P(\log(a^ib^j)), \quad i,j \in \{-1,0,1\}$$
where $P$ is a polynomial and $n \ge m \ge 1$ are natural numbers. Using Binomial identities for the ...
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Logarithms. Express $\log_3 5$ in terms of $p$ and $q$
How can I express $\log_3 5$ in terms of p and q whereby $$ p = \log_{10} 5 $$ and $$ q = \log_3 2 $$.
The given solution to this problem is $ \frac{pq}{(1 - p)} $.
Currently I'm stuck at the changing ...
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What is $\log_2{\aleph_0}$?
I understand that $\aleph_0$ is the cardinality of the natural numbers, as well as any set A, for which there’s a way to both match every element to of A to the natural numbers, and match every ...
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Does complex log(1) have other values than 0?
I want to determine all values of $$ \left[\log \left(3+2 i^{2}\right)\right]^{1-i} $$
First I simplify to $$\left[\log \left(3-2\right)\right]^{1-i}$$
resulting in $$\left[\log \left(1\right)\right]^{...
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How to solve $x+1=5e^{4x}$ [closed]
How to solve $x+1=5e^{4x}$
In general, I know to take ln() of both sides to bring down the exponent for e, but the left side is also a variable.
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Logarithmic trig equation. Why is my solution wrong?
This is the problem: $\log^2_{4}{\cos2x} = \log_{\frac{1}{16}}{\cos2x}$.
My solution:
$$\log^2_{4}{\cos2x} = -\frac{1}{2}\log_{4}{\cos2x}$$
$$\log_{4}{\cos2x}(\log_{4}{\cos2x} + \frac{1}{2}) = 0$$
$$\...
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What type of spiral is that on the picture ? and what is the formula of such?
I have found some types of spirals, and when I analysed those I have found, they do not met the criteria to shape the draw desired.
And a observation point, bacause I think spirograph its a wrong name ...
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Convergence for a sequence using logarithmic properties?
Given the sequence $x_n=\frac{\log(3n+2)}{\log(n^2+2)}$, which has the limit $\frac{1}{2}$ when n goes to infinity, give a formal proof of the limit using the epsilon definiton.
$$| x_n - L |< \...
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Equivalent functions of 8log(2(x)) [closed]
This is an assignment so I'm NOT looking for the answer but please help me understand where I went wrong.
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Have you seen this logarithmic inequality: $4\log(x)\log(y) \leq \log(xy)^2$
I have stumbled upon this following logarithmic inequality relating the product of two logs. For every $x,y > 0$
$$
4\log(x)\log(y) \leq \log(xy)^2.
$$
Furthermore, it holds as an equality if and ...
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Questions regarding logarithmic equation
This question is regarding the following problem
If
$$(y-z)^2 + (\ln(x))^2 - \ln(x)\cdot y - 2\ln(x) - z^3\ln(x) - yz^3 - 2z^3=0$$
Then find the value of
$$2y - 4 - ln(x) - z^3$$
Have been banging my ...
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Solving a logarithmic equation with different logarithmic exponents.
I had a logarithmic equation which originally was
original
https://i.sstatic.net/2fSf1jNM.png
$$5^{\log_{10}x}-3^{\log_{10}x-1}=3^{\log_{10}x+1}-5^{\log_{10}x-1}$$
but I thought that this should also ...
