Questions related to real and complex logarithms.

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2answers
27 views

Prove that from the equalities, $\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$ follows $x^yy^x=y^zz^y=z^xx^z$.

Problem : Prove that from the equalities, $$\frac{x(y+z-x)}{\log x}=\frac{y(x+z-y)}{\log y}=\frac{z(y+x-z)}{\log z}$$ follows $$x^yy^x=y^zz^y=z^xx^z$$. My approach : $$\frac{x(y+z-x)}{\log ...
7
votes
3answers
149 views

A reason for the value of $\int_{0}^{1}\log{(x)}\log{(1-x)}\,\mathrm{d}x$

In this .pdf document, which is just a list of Putnam-style undergraduate-level problems from various sources, the third question is as I have stated it below (up to a change of notation). ...
0
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3answers
41 views

Simplifying two logarithms with different bases

I am being asked to simplify: $(\log_4 7)(\log_7 5)$ How can this be simplified given that the bases are different?
0
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1answer
34 views

Logarithm of the dot product of two vectors.

Let $\vec{u}$ and $\vec{v}$ be two vectors and $\vec{u}\cdot\vec{v}$ be their dot product. My question is that how to take the logarithm of the dot product, that is, how do we find ...
0
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1answer
39 views

Partial integration of $\sin x\log(y-1)$ w.r.t. $x$

If I have the function $\sin x\log(y-1)$ and I want to partially integrate it w.r.t. $x$ then what happens to $\log $? Would the solution be: $-\cos x \log(y-1)$ and how? Isn't $\log(y-1)$ a function ...
0
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3answers
39 views

$a^{\log_g b} = b^{\log_g a}$?

$g^{\log_g a} = a$, because it equals $a^{\log_g g}$. Does this mean that $a^{\log_g b} = b^{\log_g a}$? Note: thanks whoever edited it to proper markup
0
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0answers
43 views

tricky derivative with logarithm of sum

I'm having trouble understanding the solution of a limit. It involves a formula for measuring certainty of a discrete probability distribution. Given a set of values $p_j$ which sum up to 1, find the ...
-3
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0answers
17 views

what is formula to this eqution [(256)16]1/32+[(169)6]1/12 [on hold]

how to solve this equation [(256)16]1/32+[(169)6]1/12 what is formula of this? What is the closed form expression for this? What is the right domain for this Hamiltonian 2? what is the right ...
0
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0answers
9 views

Tweaking function to reduce the rate of decay of a logarithmic based curve

Im not even sure if this is possible or perhaps I may need to use a different function altogether but I currently have one that looks like this: $$y = a\log(x+b)+c$$ That produces the red curve ...
2
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1answer
36 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
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1answer
35 views

Hard logarithm question [on hold]

$\log_a x$, $\log_b x$ and $\log_c x$ are in ap, where $x \ne 1$, then show that $$c^2=(ac)^{log_a b}.$$
3
votes
2answers
80 views

evaluating some limits with $\ln(x)$

I don't understand how to prove these results. $\lim\limits_{x \to +\infty}\dfrac{\ln{x}}{x} = 0$ $\lim\limits_{x \to 0^{+}}x\ln{x} = 0$
0
votes
2answers
77 views

How many pairs $(m, n)$ exist?

For certain pairs $ (m,n)$ of positive integers with $ m\ge n$ there are exactly $ 50$ distinct positive integers $ k$ such that $ |\log m - \log k| < \log n$. Find the sum of all possible ...
3
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1answer
25 views

Primitive of $dz/z$ is a branch of log

Let $D$ a connected open set of $\mathbb{C}$. A continuous function $f:D\to \mathbb{C}$ is a branch of log if $e^{f(t)}=t$ on $D$. In my book (Cartan) it is written that if $F$ is a primitive of the ...
14
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3answers
183 views

Prove that $\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $

I've found the following identity. $$\int_0^1 \frac{1}{1+\ln^2 x}\,dx = \int_1^\infty \frac{\sin(x-1)}{x}\,dx $$ I could verify it by using CAS, and calculate the integrals in term of ...
1
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1answer
82 views

Solve $\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$ for $a/b$

The question: $$\log_9 (a) + \log_{12} (b) = \log_{16} (a+b)$$ solve for $a/b$. It gives hints: put it all in terms of x. $$9^x=a$$ $$12^x=b$$ $$16^x=a+b$$ Now prove that: $b^2=a(a+b)$ I did and ...
0
votes
1answer
29 views

A question to do with logarithms?

$\log_3x^3 + {3\over \log_3x} =4$ Ok, the way the computer has put it makes it look weird.But it is :log to base 3 of "x" to power 3 plus 3 divided by log to base 3 of x is equals to "4". This ...
3
votes
3answers
48 views

Solving logarithmic equations

The equation that I'm trying to solve is: $$\log _{5x+9}(x^2+6x+9)+\log _{x+3}(5x^2+24x+27)=4$$ Using algebra and principles of logarithms I managed to get the equation down to $$\frac{2\left(\log ...
0
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0answers
15 views

Help finding a formula to fit the data - both axes are logarithmic?

I'm an electrical engineer trying to come up with a formula to turn a measurement of light level into lux. The photoresistor changes resistance logarithmically. Lux is also logarithmic. So the data I ...
4
votes
4answers
77 views

Is $\ln(x^{p(x)}) = p(x) \ln(x)$?

I am trying to prove that: $x^{\frac{\ln(\ln(x))}{\ln(x)}} = \ln(x)$ My "solution": $e^{\ln\left(x^{\frac{\ln(\ln(x))}{\ln(x)}}\right)} = e^{\frac{\ln(\ln(x))}{\ln(x)} \ln(x)} = e^{\ln(\ln(x))} = ...
4
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1answer
61 views

Is there another function with a property like the log?

Is there another differentiable monotone increasing (or decreasing) function $ f:\mathbb{R} \rightarrow \mathbb{R} $ with a property that $ f(xy) = f(x) + f(y) $, like the log-function has it?
2
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6answers
36 views

Solving for $x$ in an exponential equation

Say we the following equation $$F(x) = \frac{\exp(a+bx)}{1 + \exp(a+bx)}$$ Now we set $x=0$ and we want to solve for $a$ as a function of $F_0$. So that, we have: $$F_0 = \frac{\exp(a)}{1 + ...
2
votes
1answer
48 views

Logarithm question, does $\ln \sqrt{7}$ equal to zero?

I got this from my workbook solution, was able to solve the question for the most part but stuck in the last sentence. $$7(\ln\left|x+\sqrt{x^2-7}\right|-\ln\sqrt{7}) + c = 7 ...
0
votes
2answers
29 views

Show using logarithms that the first equation can be transformed into the second.

Show using logarithms that if $y^k = (1-k)zx^k(a)^{-1}$ then $y = (1-k)^{(1/k)}z^{(1/k)}x(a)^{(-1/k)}$.
9
votes
5answers
547 views

How to solve an exponential and logarithmic system of equations?

$$ \left\{\begin{array}{c} e^{2x} + e^y = 800 \\ 3\ln(x) + \ln(y) = 5 \end{array}\right.$$ I understand how to solve system of equations, logarithmic rules, and the fact that $\ln(e^x) = e^{\ln(x)} ...
1
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2answers
93 views

Solving $x^{2n} = \frac{1}{2^n}$ for $x$

What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation $$x^{2n} = \frac{1}{2^n}$$ for ...
1
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3answers
39 views

How to find range of a logarithmic function?

How do I find the range of these logarithmic functions? \begin{align} & \ln(3x^2 -4x +5), \\ & \log_3(5+4x-x^2). \end{align} how should I approach questions like this ? What I did: I found ...
1
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3answers
41 views

Proving: $\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0.$

$$\begin{equation}\frac{2x}{2+x}\le \ln (1+x)\le \frac{x}{2}\frac{2+x}{1+x}, \quad x>0\end{equation}$$ I found this inequality in this paper: http://ajmaa.org/RGMIA/papers/v7n2/pade.pdf (Equation ...
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1answer
32 views

Natural Logarithms (Help) [closed]

Can I have an answer with working out to this question please, I am puzzled at the result I got and don't believe it is right. The question asks to 'Simplify by expressing as a single natural ...
0
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0answers
21 views

Manipulating product of two matrices

In a published paper I saw the following $$\log \left(\mathbf{I} + \mathbf{T}\mathbf {Hpp^HH^H}\right)= \log(1+\mathbf {p^HH^HTHp})$$ where uppercase means a matrix while lower case means vector ...
2
votes
1answer
68 views

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$?

Is $f(\sqrt{x}) = \sqrt{f(x)}$ true, where $f(xy) = f(x) + f(y)$ for all positives $x, y \in \mathbb{R}$? It's obviously false. But the point is that "can it be proved without using the fact that the ...
0
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1answer
31 views

When does function $(\log_b(x))^p$ change its curvature?

Consider $(\log_b(x))^p$ where $b$ is a constant $>1$; $x, p \in \mathbb R_+$. As we increase the value of $p$ (starting from 1), at specific value of $p$, the curve changes its shape from ...
1
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1answer
31 views

Taking Log to find MAXIMIZE summation of variables

I have been reading IEEE papers on communication and in several papers the authors formed objective function like: $\text{Maximize } \sum_k \log r_k $ to maximize the total rate of the system of ...
0
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0answers
19 views

What's the non-log form of this equation?

I found this equation in a book and I'd like to know what it would look like in 'non-log' form: $(a \log x + b \log N)^2 + 2g \log x + 2f \log N + C = 0$ Thanks.
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4answers
55 views

Integrating for a solution in terms of an natural logarithm

Evaluating the following integral: $$\int_1^2 \frac2{1-3x}\ dx$$ why do you have to take the factor of $-2/3$ out when evaluating the integral?
1
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0answers
41 views

solve $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$

Is it possible to find the analytical solution of $a\cdot e^{b\cdot x}+c\cdot ln(x)=0$? Is that a transcendental equation?
2
votes
6answers
85 views

How do I solve for $t$ in this equation?

I know I'm supposed to use $\ln()$ to work it out, but I can't remember how it's done. Can anyone help? The equation is $$ 40e^{-t/5}=20 $$
1
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2answers
32 views

Does the matrix logarithm always converge for exponential matrices?

We have defined functions on square matrices $X$: $$e^X := I + X + \dfrac{X^2}{2!}+\dfrac{X^3}{3!}$$ and $$log(X):= (X-I)-\dfrac{(X-I)^2}{2} + \dfrac{(X-I)^3}{3}-...$$ The exponential converges for ...
3
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2answers
78 views

Evaluation of $\int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx =-\frac{\pi^4}{15}$ and $\int_{-\pi}^{\pi} \log(2\cos{\frac{x}{2}}) dx =0$

In the following encyclopedia, http://m.encyclopedia-of-equation.webnode.jp/including-integral/ I found the relations below \begin{eqnarray} \int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx ...
0
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2answers
63 views

Approximation of the Gamma function

I am having trouble obtaining a lower bound for the following formula: $$ \ln\frac{\Gamma\left(\frac{x}{3}\right)}{\Gamma\left(\frac{x}{4}+1\right)\Gamma\left(\frac{x}{12}+1\right)}. $$ I tried using ...
0
votes
2answers
30 views

Find the value of x if 1,$\log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ are in an AP.

We have an AP: 1, $ \log_{9}(3^{1-x}+2) $, $ \log_{3}(4.3^x-1) $ We have to find value of x. $$ d = a_{2} - a_{1} $$ $$ d = log^{(3^{1-x}+2)}_9 - 1 $$ $$ d = log^{(3^{1-x}+2)^{\frac{1}{2}}}_3 - ...
0
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0answers
20 views

solve implicit equation with lambertw, exponentials, logarithms and first order polynom

I have a very complicated problem to solve. I am almost sure it's impossible to solve but maybe one of you guys has a miracle solution for me. I am modelling the behaviour of a photovoltaic cell and ...
1
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2answers
44 views

Is the function $\ln(ax + b)$ increasing/decreasing, concave/convex?

$h(x) = \ln(ax + b)$ NB. Examine your results acccording to values of $(a,b)$ I've differentiated twice in order to get the following: $$ h''(x) = -\frac{a^2}{(ax+b)^2} $$ I think this proves ...
0
votes
5answers
109 views

How to resolve $n>(1+\frac{1}{n})^n$?

I'm trying to prove that $\forall n\geq 3, n^{n+1}>(n+1)^n$. I came that this is true for $n>(1+\frac{1}{n})^n$. WolphramAlpha gives $n>2.293166...$ but I failed to compute it analytically.
0
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1answer
46 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
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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!