Questions related to real and complex logarithms.

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24
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1answer
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What is a closed form of $\int_0^1\ln(-\ln x)\ \text{li}\ x\ dx$

Let $\operatorname{li} x$ denote the logarithmic integral: $$\operatorname{li} x=\int_0^x\frac{dt}{\ln t}.$$ Is it possible to find a closed form of the following integral? $$\int_0^1\ln(-\ln x) ...
1
vote
1answer
16 views

Analyze the Complex Function by using the Principal log Branch

I am trying to analyze the function $\sqrt{1-z^2}$, where the square root function is defined by the principal branch of the log function. I want to locate the the discontinuities. I know the ...
8
votes
3answers
231 views

Integral $\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$

It's a follow-up to my previous question. Can we find an anti-derivative $$\int\arcsin x\cdot\ln^3x\,dx$$ or, at least, evaluate the definite integral $$\int_0^{1/2}\arcsin x\cdot\ln^3x\,dx$$ in a ...
0
votes
0answers
69 views

The domain of $\frac{\ln x}{x}$.

I have to find the domain for $f(x) = \frac{\ln x}{x}$ Naturally, $x$ must be larger than $0$ and $x$ can't be $0$ so $x > 0$. But when I graphed the function, it has two "parts", one in the ...
4
votes
1answer
196 views

Proof that $\frac{2}{3} < \log(2) < \frac{7}{10}$

Positive integrals $$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$ and $$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ (http://math.stackexchange.com/a/1618454/134791) prove that ...
5
votes
7answers
203 views

Prove that $\frac{7}{12}<\ln 2<\frac{5}{6}$ using real analysis

I studying in Real Analysis 2, but I have no idea how to solve this problem. My guess is to use Mean Value Theorem or a similar theorem? Could any one help me? Thanks.
-1
votes
2answers
69 views

To evaluate $\lim_{x \to 0^+} \frac{\log(x)}{\sqrt x}$ using inequality [on hold]

To evaluate $$\lim_{x \to 0^+} \frac{\log(x)}{\sqrt x}$$ I know that $\log(x) < x$ for $x > 0$. So dividing by square root of $x$ and taking limits gives me nothing. Which inequality should I ...
0
votes
1answer
24 views

Is $\log (t^2 (l/c)) = \log (t^2) \log (l/c)$?

I'm new in this forum want to ask a beginner question about logarithm: Is $\log (t^2 (l/c)) = \log (t^2) \log (l/c)$?
0
votes
0answers
23 views

Help with understanding when Log(z^k)=k Log(z) as well as drawing the function.

For the question I'm dealing with the property Log(z^k)=k*Log(z)in which I have to find the largest open set that this property is true when $k$ is a positive integer. I understand that this ...
3
votes
0answers
55 views

Want to know what's wrong?

I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ...
1
vote
4answers
53 views

Find the limit of fraction involving logarithms

I am looking for a way to prove the following limit for integer $x$s: $$\lim_{x\to\infty}{\frac{\log(x+2)-\log(x+1)}{\log(x+2)-\log(x)}}=\frac{1}{2}$$ I could find the result by using a computer ...
2
votes
2answers
34 views

Is there an alternative approach to the definition of prime numbers based on the definition of the natural logarithm?

Is there an alternative approach to the definition of prime numbers based on the definition of the natural logarithm? These are my thoughts about it, the questions are at the end: Basically when a ...
0
votes
0answers
11 views

how would zipf's law be expressed as a logarithmic function?

I am doing a project for math class based on Zipf's law but i cannot understand how it relates to logarithms. The project handout states that "In 1949, George Zipf noticed that if you tabulate the ...
-1
votes
1answer
48 views

How do I solve logarithms with addition in them? [on hold]

I've asked wolfram alpha to see if there is any solution and in fact there is. Now I just need a way to do this by hand because my school wants me to know such things. So I would really appreciate any ...
0
votes
2answers
30 views

Expanding logarithm of function

Is there a way (there has to be), I can expand an expression like this? $$\log_2 (3f(n)^n)$$ P.S. This part of an assignment I'm working on, please do not give solutions
-4
votes
0answers
54 views

solve $4x^2y'' + y=0$, $y(-1)=2, y'(-1)=4$

This would require taking the $\ln(-1)$, which Zill solved in the 7th edition of diff eq $4.7$ problem $37$ by substituting $t$ for $x, y(1)=2, y'(1)=4$. Then substituting $-x$ for $t$ in the final ...
1
vote
2answers
40 views

Find all $x$ such that $8^x(3x+1)=4$

Find all $x$ such that $8^x(3x+1)=4$,and prove that you have found all values of $x$ that satisfy this equation. My effort Rewriting the equation I have \begin{array} 22^{3x}(3x+1)&=2^2 \\ ...
4
votes
3answers
434 views

Find root of the equation

Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$ It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
2
votes
2answers
28 views

How to express $\log_5 2$ in terms of a and b (Refer to qn)

In my textbook, I came across this interesting question which I am currently struggling to solve: If $\log_6 2 = a$ and $\log_5 3 = b$, express $\log_5 2$ in terms of a and b The solution given is ...
0
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1answer
30 views

Prove that $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$

Show that if $n$ is a power of $2$, say $n = 2^k$, then we have the equality $\sum_{i=0}^{k} \lg \frac{n}{2^i} = \Theta(\lg^2 n)$. The first step is to prove $O(\lg^2n)$: $$ \lg \frac{2^k}{2^0} + \lg ...
1
vote
3answers
63 views

Prove $\ln x \ge \frac{x-1}{x}$

Prove that for every $x>0$: $$\ln x \ge \frac{x-1}{x}$$ What I did: $$f(x) = \ln x, \text{ } g(x) = \frac{x-1}{x} $$ $$f(1) = g(1) = 0 $$ So it's enough to prove that $$ f'(x) \ge g'(x)$$ ...
0
votes
5answers
41 views

Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

Equation: $$\log(x^2+2ax)=\log(4x-4a-13)$$ It has only one solution; then exhaustive set of values of $a$ is ?? I don't even know where to begin The answer is : $$(-13/4,-13/12) \cup [-1]$$
0
votes
1answer
31 views

Showing that $\log{\log^d{3n}} = O(\log{\log^d{n}})$

I'm trying to show this: $$\log{\log^d{3n}} \leq q\cdot \log{\log^d{n}} \;\;\exists\, q,k > 0,\forall n>k, \text{where } d \text{ is a constant} > 0$$ This is what I have so far ...
3
votes
1answer
75 views

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$?

Is it possible to solve $k = \frac{x}{\ln(x)}$ for $x$? My suspicion after a fruitless hour of manipulation is that it is not.
0
votes
1answer
26 views

Solving for numerator in equation with logarithms (Activation Energy Equation)

I'm having trouble solving for k1 in this equation: ln(0.286/k1) = (100000/8.314)(1/500 - 1/490) The right side should equal 0.491, which I can calculate just fine, but then the left side gives me ...
0
votes
1answer
29 views

Converting equation to slope-intercept form

It's been awhile since I've worked problems like these and I am a bit hazy on some of the rules. I was hoping someone could show me how these are solved so that I can make sure I'm on the right path: ...
0
votes
2answers
25 views

Most logical thing to do with these exponents and sums?

I'm doing homework for a programming class and came across this problem. There's no directions besides what I've shown, so I don't even know what it's asking me to do. What makes the most sense for ...
1
vote
1answer
40 views

Closed-form Solution of Log Sum

I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ ...
0
votes
0answers
9 views

Calculating Data Rate using Quadrature Amplitude Modulation (QAM)

I was working on my telecommunications homework and I have these questions: Calculate the data rate for a 2400 baud signal where each symbol can take on one of two levels (M=2) Calculate the data ...
4
votes
1answer
73 views

How to solve $\ln(y)=\ln(x)e^{\ln(x+1)} $ for x?

I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do $y = x^x$ // $x = e^{\ln(x)}$ $y=x^{e^{\ln(x)}}$ // $\ln$() $\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I ...
0
votes
1answer
24 views

solve and skecth $\log{|z|}=-2\arg(z)$

Ive asked this question a week ago, but nobody managed to answer but it is doing my heading from then. I know usually You demand some initial work done on the question but I just dont know how to ...
0
votes
2answers
27 views

Solving an exponential equation with x as a base and an exponent

So here's the problem: $x+3=3^x$ Obviously, graphing both sides and finding the intersection would reveal the answer, but algebraically, how can this be solved?
0
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0answers
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Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...
0
votes
3answers
62 views

$\frac{\ln(x^2)}{\ln(x)} = 2$? Why?

$\frac{\ln(x^2)}{\ln(x)} = 2$? Upon trying to evaluate $\frac{\ln(x^2)}{\ln(x)}$, i've found that google plots it as always equal to 2, other than 0 where it is undefined. Why is this the case?
1
vote
2answers
31 views

Order of growth of logarithms, compared to linear

I think it is true that any power of a logarithm, no matter how big, will eventually grow slower than a linear function with positive slope. Is it true that for any exponent $m>0$ (no matter how ...
1
vote
1answer
53 views

How do you differentiate the integral from $ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt$ [duplicate]

How do you differentiate the integral from $e^{-x}$ to $e^x$ of $\sqrt(1+t^2)$ with respect to t? $$ \int_{e^{-x}}^{e^x} \sqrt{1+t^2}\,dt $$ I know the answer is $$ e^x\sqrt{1+e^{2x}} + ...
0
votes
1answer
42 views

Does $\log_2 \sqrt[4]4$ exist?

Tomorrow I have an exam about graphics and log operations. Our teacher gave us a paper with exercises to practice and one of the exercises is: $\log_2 \sqrt[4]4$ I couldn't find the solution. ...
1
vote
1answer
33 views

Derivative of matrix logarithm with respect to matrix

I saw in this post that $\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$ Is this true to say: $\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$ where U is ...
2
votes
2answers
34 views

Solve for $x\quad \log_2(2^n) = \log_2(1+x)$

I am out of practice with logs, but this is derived from the channel capacity theorem. $$B\log_2\left(1 + \frac SN\right)$$ Solve for $x $ $$\log_2(2^n) = \log_2(1+x)$$ I need this equation ...
0
votes
3answers
114 views

Exponential (to the power of a logarithm) [closed]

How do I solve the following equation: $(3x)^{ln3}=(4x)^{ln4}$ Thanks in advance!
0
votes
1answer
38 views

Prove that $\sum{\log_{ab^2c^2}a = \frac35}$ only if $a=b=c$ [closed]

So I have $a,b,c \in (1, \infty)$. Prove that $$ \log_{ab^2c^2}a + \log_{a^2bc^2}b + \log_{a^2b^2c}c = \frac35 \Leftrightarrow a=b=c $$
0
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3answers
54 views

Show that $\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$

Prove the following: $$\frac{1}{1+k}=\frac{\frac{1}{k}}{1+\frac{1}{k}}\leq \ln(1+\frac{1}{k})\leq\frac{1}{k}$$ I know I can prove it with induction if the values were naturals. However, the "problem" ...
3
votes
3answers
120 views

Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$

I want to show that for all $a \in \mathbb{R }$ $$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$ So far I've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
2
votes
1answer
33 views

Stuck with understanding transformation step in calculating limit of $n(\sqrt[n]{a}-1)$

Although this question has already been asked in general ( $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$) , my question is different, because I am stuck with a specific transformation step: ...
1
vote
3answers
65 views

Proof of the fact that $\ln(a) = f '(0)$ for $f(x) = a^x$?

Looking over notes from class today and wanted to know if there is any type of proof for the fact that $\ln(a) = \lim_{h\to0}(a^h-1)/h$, which is just $f '(0)$ for any function of the form $f(x) = ...
0
votes
1answer
33 views

How to show $\log a\le n(\sqrt[n]{a}-1) \le \sqrt[n]{a}\log a$

Let $ b=\sqrt[n]{a}$. How to show: $\log a\le n(\sqrt[n]{a}-1) \le \sqrt[n]{a}\log a$? Thank you ;)
3
votes
1answer
126 views

Intuitive understanding of logarithms

I know logarithms are supposed to be the inverse of exponential functions, and while this makes sense, it seems to me that a more intuitive and significant property is $$\log (ab) = \log(a)+\log(b)$$ ...
0
votes
2answers
25 views

Difficulty finding the sum of a hyperbolic function.

Can someone please point out where I am (If I am) going wrong during the solution process of the following question: I have been presented with the following : $$4sinh(2ln(2))-cosh(ln2)$$ and told ...
1
vote
2answers
43 views

taking the natural log of e^(2x) = (4/3)

I have been unable to answer the following question. I must solve for x: $$e^{2x} = (4/3)$$ I have been made aware that I must take the natural log of both sides, giving: $$ln(e^{2x}) = ln(4/3)$$ ...
3
votes
3answers
188 views

How to solve $\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$?

Here is my question $$\int_0^{\infty} \frac{\log(x+\frac{1}{x})}{1+x^2}dx$$ I have tried it by substituting $x$ = $\frac{1}{t}$. I got the answer $0$ but the correct answer is $\pi log(2)$. Any ...