Questions related to real and complex logarithms.

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0
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0answers
20 views

Computing the Log-Euclidean distance efficiently by using eigen-analysis.

Let $A,B\in\Bbb{S}_{++}^n$ be two symmetric positive definite $n\times n$ matrices with real entries. The Log-Euclidean distance between these matrices is defined as follows $$ d = \lVert \log(A) - ...
2
votes
1answer
37 views

$\log (A + \delta A) = ?$ (as an expansion in $\delta A$), where $A$ and $\delta A $ are matrices

$A$ and $\delta A$ are two non-commuting matrices and I am seeking a power series expansion to 2nd order in $\delta A$. After writing it as $\log (A (1 + A^{-1}\delta A) )$, I am unable to figure out ...
0
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2answers
51 views

Integral of logarithm of exponential function

I am trying to solve this integral: $$\int \log\left(1 + \frac{1}{\pi}\exp\left(\frac{-x^2}{2a^2}\right)\right) dx$$ where $a$ is some fixed constant. The bounds of this integral are $-a$ and $a$, ...
0
votes
2answers
39 views

Maximum number of digits in numbers between 0 to $n^2-1$ of base n

The number of digits in numbers between 0 and $n^2-1$ of base n is obtained by $\log_n(n^2) = 2\log_nn = 2$ But why log is ...
1
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1answer
35 views

Integral formula involving logarithms and the zeros of a holomorphic function

I have the following formula I´d like to prove: Given a holomorphic function $f:U\to \mathbb C$ such that $\overline{D_r(0)}\subset U$, $f(0)\neq 0$ and $f(z)\neq 0$ for $z\in \partial D_r(0)$, we ...
0
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2answers
83 views

Find x, if $ \log _{15}\left(\frac{2}{9}\right)^{\:}=\log _3\left(x\right)=\log _5\left(1-x\right) $

So how can I find the value of x, if: $$ \log _{15}\left(\frac{2}{9}\right)^{\:}=\log _3\left(x\right)=\log _5\left(1-x\right) $$ I tried switching everything to base 15, but that didn't work out ...
3
votes
1answer
53 views

How to find the center of a log spiral?

Given just a few points on a log spiral, how to find the center? Considering that the circle is a degenerate case of the log spiral, is there a way to generalize the method for finding circle centers ...
-1
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2answers
34 views

Log of negative numbers

I know that log of negative numbers is complex numbers. But I just got over this little proof and wondering what is wrong with this? ...
2
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2answers
25 views

$ 2\log ^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0$

Find the sum of solutions to: $$ 2\log^2_{4}(|x+1|)+\log_4(|x^2-1|)+\log_{\frac{1}{4}}(|x-1|)=0 $$ I'm not sure about what to do with the absolute values, how can I get rid of them? Should I solve ...
3
votes
3answers
57 views

How to prove that $a^{\log_cb}=b^{\log_ca}$

I've met a question whereby it asked me to show that $a^{\log_cb}=b^{\log_ca}$. I'm okay with the other logarithm questions. But I don't know how to show this question out. Can anyone give some hints ...
1
vote
1answer
35 views

If log8n=1/2p, log22n=q, and q-p=4, find n [duplicate]

I'm having a hard time finding the value of $a$ in this problem. My teacher was trying to explain to me the process in which to get it but I did not understand him.
2
votes
2answers
61 views

Evaluate the limit: $ \lim_{x\to -1}\frac{x\ln(x+3) + \ln(2)} {x+1} $

$$ \lim_{x\to -1}\frac{x\ln(x+3) + \ln(2)} {x+1} $$ I tried to separate the fraction and also a change of variable (x+3 = y+1) but I couldn't solve it. Maybe there's a trivial step that I'm just ...
0
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2answers
53 views

Where is my mistake in a logarithm?

Prove that $$3^{\log_2 5} = 5^{\log_2 3}$$ is true. Here is my solution:
-4
votes
3answers
57 views

How to prove that: $\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$ [closed]

Prove that: $$\log_{{1\over 2}}(3) + \log_3\left({1 \over 2}\right) < -2$$ Please help me solve it.
0
votes
2answers
44 views

Product of all real solutions of equation $\frac {2013x}{2014}=2013^{\log_x2014}$?

How am I even supposed to start this task, i need some hint? I logarithm both sides and these are my steps: $$\frac{2013x}{2014}=2013^{\log_x2014}$$ ...
1
vote
1answer
42 views

Finding the limit of $\lim_{n\to\infty} \frac{n^{log(n)}}{(\log n)^n}$

I try to calculate the following limit: $$\lim_{n\to\infty} \frac{n^{\log(n)}}{(\log n)^n}$$ I tried this: $$ \lim_{n\to\infty} e^{(\log(n))^2 - n \log(\log(n))} $$ Is this useful? & what ...
0
votes
2answers
26 views

If $x^y = y^x$ $(x,y \in R, x,y>0,x\neq 0 )$ and $x^p = y^q$ $(p,q \in R/\{0\}, p \neq q)$, then product $xy$ is equal to?

Solution for this one is $({\frac{p}{q}})^{\frac {p+q}{p-q}}$ , but I do not understand how I am supposed to get here, I guess something with logarithms but not sure what?
1
vote
2answers
52 views

Why is the derivative of $\log ax$, where a is any positive integer, the same?

For a question in my textbook: Differentiate $\log(2x)$ The differentiation rule for logarithm is $1/x \ln b$, where $b$ is the base. So my answer was $1/(2x) \ln 10$, but the answer my textbook ...
2
votes
4answers
40 views

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$.

Find the value of $x$ such that $(3-\log_3x)\log _{3x}3=1$. Is there another way to solve other than this attempt? My attempt, $(3-\log_3x)\log _{3x}3=1$ $\frac{\log(3)\left(3-\frac{\log (x)}{\log ...
1
vote
1answer
59 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
0
votes
2answers
37 views

If $\log_23 = a$ and $\log_52=b$ then $\log_{24}50$ is equal to?

I guess this has to be done by using simple logarithmic rules, but I do not how to start. Answer in my booklet is ${b+2}\over{b(a+3)}$
0
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2answers
43 views

What is the sum of $n$ terms in the series

What is the sum of $n$ terms in the series : $$\log m+\log \left(\dfrac{m^2}{n}\right) +\log \left(\dfrac{m^4}{n^3}\right) +\cdots\cdots$$ Options $ a.)\ \log ...
2
votes
3answers
43 views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

I could prove it by induction, but I need to see how I might have discovered it for myself (cause that's what's gonna be on exam).
0
votes
2answers
39 views

All real number solutions of equation $\log_{2011}(2010x) = \log_{2010}(2011x)$ are in certain interval. Which one is it?

This task has to be done with no calculator, but I don't have basic idea how to start. Can someone give me advice, I know this is pretty easy but I need direction for particularly this one? EDIT: I do ...
2
votes
3answers
63 views

Value of $\frac{1}{\log_aabc}+\frac{1}{\log_babc}+\frac{1}{\log_cabc}$

How to find the value of $\frac{1}{\log_aabc}+\frac{1}{\log_babc}+\frac{1}{\log_cabc}$? I guess the answer will be $1$. But I don't know how to evaluate it. Can someone give me some tips?
1
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2answers
25 views

Assuming $d+1 <= log_2(n)$, how to show $d - 1 > log_2(n/8)$?

Also we know $d = log_2(n/2)$ rounded down to its nearest integer. Add (-2) to each side $$d-1 <= log_2(n) - 2$$ $$d-1 <= log_2(n) - log_2(4)$$ $$d-1 <= log_2(n/4)$$ This is as far as I can ...
4
votes
3answers
174 views

$a=b^x+c^x$, How to solve for $x$?

If $a=b^x$, then $x$ could be written in terms of $a$ and $b$; $x=\dfrac{\log(a)}{\log(b)}$. What about $a=b^x+c^x$? Could $x$ be written in terms of $a, b$ and $c$? $x={}$?
3
votes
4answers
148 views

How to compute the derivative of $\sqrt{x}^{\sqrt{x}}$?

I know have the final answer and know I need to use the natural log but I'm confused about why that is. Could someone walk through it step by step?
2
votes
2answers
41 views

Simplifying the equation $\log y = 10 + 0.5x$

Solve for $y$. When expressed in simplest form, what familiar kind of equation results? $$\log y = 10 + 0.5x$$ For this question, I would get rid of log first right? So, I would get ...
0
votes
2answers
29 views

Comparing the greatest values of two functions (Derivatives)

I've tried doing this task, and for this kind of task I should be using derivatives. When I done all the calculus, everything I got were some weird result which I do not know how to compare. Task ...
0
votes
1answer
54 views

What's the derivative of $\ln \lvert \csc x + \cot x \rvert$? [closed]

What is the derivative of $\ln \lvert \csc x + \cot x \rvert$? I've tried to do it and I get really odd numbers, any help showing the steps would be very helpful!
0
votes
0answers
15 views

Calculating vth moment of log-normal pdf describing particle size distribution

The number of density particles having radii between $r$ and $r+dr$, is given by the equation: $$n(r)=\frac{1}{\sqrt{2π} r \ln(σ_x)} e^{-\frac{ln^2(r/r_g)}{2ln^2(\sigma_x)}}$$ where $\sigma_x$ is ...
0
votes
2answers
27 views

Multi-valued logarithmic function

I'm reading some notes for an electrical engineering class and came to the following: "...$2^j$ can represent a countably infinite number of real numbers. These examples are related to the fact that ...
2
votes
0answers
23 views

Bias induced by splitting a log sum into independent log [closed]

Does anyone know the introduced bias ($\epsilon$) when a log-sum is split into a sum-log $$\log \left(\sum_{k=1}^N a_k\right) = \sum_{k=1}^N \log(a_k) + \epsilon$$ Many thanks for your help!
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votes
3answers
45 views

Differentiation using logarithms.

the variables $x$ and $y$ are positive and related by $$x^a\cdot y^b=(x+y)^{(a+b)}$$ where $a$ and $b$ are positive constants. By taking logarithms of both sides, show that ...
3
votes
1answer
99 views

Sum of infinite series and ratio of powers.

If $A = 1 + r^{a} + r^{2a} + \cdots ~~~~~ ;~~~ B= 1 + r^{b} + r^{2b} + \cdots $ Then $a/b$ is ? I took the sum of both series : $$A = \dfrac{1}{1-r^a}~~;~~~ B=\dfrac{1}{1-r^b}$$ But now how do we ...
7
votes
2answers
559 views

Is there ANY possible way to solve this equation?

So I came up with this equation and it just seems like I can't solve it AT ALL for '$a$' $$a*b^a = c$$ EDIT: By the way, I'm only taking $b^a$, not both $b$ and $a$, just in case anyone was ...
3
votes
6answers
133 views

Is there proof show that $\log x$ is undefined and make no sense at $ x=0$?

I was asked by someone: why $\log x$ is undefined at $x=0 $? Is there proof show that $\log x$ is undefined at $x=0$? Note(01):: log is the inverse function of the exponential function. note(02): ...
1
vote
3answers
70 views

How do I prove $\log(x^n)=n\log|x|$?

By definition we know that: $\log(x^n)=n\log|x|$ as known property in logarithm function . If it's not a trivial question, how do I prove that :$\log(x^n)=n\log|x|$? Note: $x$ is real number, $n$ is ...
1
vote
3answers
57 views

Solving the exponent function for X

Natural logarithm is defined as: $\ln(Y) = x$ Which can be also written as: $e^x = y$ Now the problem is, to solve the above equation for x you would need to use logarithm, unless the base can be ...
0
votes
2answers
43 views

I need help solving a logarithm equation.

these were my steps. Can someone tell me where have i goe wrong since the answer is $-{\frac 14}$. $\sqrt{\log(\sqrt{10}a)} = {\frac 12}$ ${\frac 12}{\log(\sqrt{10}a)} = \log\sqrt{10}$ ...
1
vote
2answers
20 views

Problem in substitution

I have a very stupid question, it seems that I've forgotten most of my math and can't figure this out. Considering the following, ...
0
votes
2answers
78 views

What is $\text{log}(-x)$?

I am having some confusion in regards to the log based value of a negative number. I know that this is said to be undefined, though I accidentally entered in '$\log(-x)$' instead of '$\log(x)$' via a ...
3
votes
1answer
56 views

$\ln r+\ln q=kr$ Isolating $r$

A problem I'm working on requires me to solve $\ln r+\ln q=kr$ for $r$. I've tried using the Lambert $W$ function, but I'm not sure how to do it. Is there method, technique or known solution, that ...
1
vote
1answer
54 views

Logarithm doubt …

I know that log of a negative number is not possible but, $\log(-5)^2$ is possible. Therefore $\log(-5)^2=2\log(-5)$ but $\log(-5)$ is not possible but $log$ of $-5$ square is possible ....can anyone ...
0
votes
1answer
52 views

Why is the function $\operatorname{Log}(G(t))$ Holder continuous?

I was reading the theory about the Riemann-Hilbert problem $\Phi^+(t)=G(t)\Phi^-(t)$ where $G(t)$ is a Holder continuous function on a closed curve $c$ with index $\operatorname{Ind}_cG(t)=0$. To ...
2
votes
4answers
367 views

Is there any way to prove this without logarithms?

I was given this problem: Show if $a>1$ and $n>1$ ($n$ and $a$ are integers) then, $\lim_{n\to\infty}a^{\frac{1}{n}}=1$ . The obvious solution is the following: Take the logarithm in base ...
0
votes
2answers
88 views

Transcendental numbers & logarithms

Given two coprime positive integers greater than one, say $n,\ m$ , where $n > m$ . How do we find the ratio $\dfrac{\log m}{\log n}$ in terms of $n$ and $m$ symbolically ? Claim: The ratio is ...
4
votes
1answer
50 views

Iteration of $\log(z) / \sqrt{z}$

The complex function $\log(z) / \sqrt{z}$ is a curiosity that I find interesting since one can express $e^{i\pi}+1=0$ as $\log(-1) / \sqrt{-1} = \pi$. My question is, what is the significance of the ...
0
votes
1answer
50 views

solve logarithmic equation without numerical methods

Is there algebraic method to solve following equation for $x$: $$ a x + b \ln x + c = 0 $$ with $a , b , c$ constants without using numerical methods and ln means natural logarithm.