Logarithm Definition and 335 Threads

Where does this definition come from: $$\ln n = \int_{0}^{\infty} \int_{1}^{n} e^{-xt} dx dt$$ Thank you very much.

are they equal?

Can anyone verify or correct this equation to work out the crossover frequencies of a three tier sound system please? I never was the best in my physics classes... (Please find attached "Crossover Freq.dpf") And thank you in advance for your time. Sorry about the way i have had to write...

Homework Statement Prove that $$log_{b}(xy)=log_{b}x+log_{b}y.$$ Homework Equations Let $$b^{u}=x,b^{v}=y.$$ Then $$log_{b}x=u,log_{b}y=v.$$ The Attempt at a Solution I'm afraid I've been using circular reasoning to prove this. I can get this to a point where I have...

The problem statement ln(x) = 5 -x Solve for x. The attempt at a solution ln x = 5 - x \\ e^{ln x} = e^{5 - x} \\ e^{ln x} = \frac{e^5}{e^x} \\ x e^x = e^5 Here is the place where I get stuck.

Does there exist a closed form formula for in(x)?

In Euclidean geometry (presumably also in non-Euclidean geometry), the part of the dissecting line that dissects the vertex angle and is inside the isosceles triangle is shorter than the legs of the isosceles triangle. Let ABC be an isosceles triangle with AB being the base. Then, for...

Hello, I know that a negative logarithm is undefined. But I am faced with an equation like this: -1 x (-2)^(n-1) = -16777216 I divided it by -1 to give (-2)^(n-1) = 16777216 And then took logs to get (n-1) log(-2) = log 16777216 Since I can't work out the log of a negative number, what do I...

Working on a personal music project, I would like to pull 'z' out of the logarithm below if I can help it, but am having trouble: It's a portion taken from this: Which evolved from... However, this is only the mathematical modeling of the object. The actual *physics* aspect...

Homework Statement Consider the equation: 3logx5+2logx2-log1/x2=3 a)State which values of x for which the equation is defined. b)Solve the equation for x. Homework EquationsThe Attempt at a Solution 3logx5+2logx2-log1/x2=3 =logx53+logx22-log1/x2=3 =logx125+logx4-log1/x2=3 =logx500-log1/x2=3...

Homework Statement A vehicle purchase for $32,000 depreciates at a rate of 75% every 6 years. Another vehicle purchased for $16,000 depreciates at a rate of 50% every 4 years. Create an exponential function for each situation, and use the functions to algebraically determine the amount of time...

Homework Statement log39x4 - log3(3x)2 The answer sheet says that the answer is 0, but when I work the problem both ways I get: 2log3x Any ideas? Homework Equations logxy/z = logxy - logxz The Attempt at a Solution Formula Sheet 20-30 Minutes of Messing around with the problem

First, I'll note this interesting thread: #4158384. However I unfortunately don't have privileges to post there so I'll start this new one. I'm trying to make my own sliderule. How did they do it? How did Napier invent his "bones?" A more particular question would be: If I make a table of...

Homework Statement Given sin x = (eix - e-ix) / 2i, I want to prove that arcsin x = -i ln(ix + √1 - x2) Homework Equations I know about the Euler's formula and complex number. But I have never learned about complex logarithms. The Attempt at a Solution I try to use x = sin y. But it seems...

I need to solve log3(2x+3)-log3(x-2)=3 where 3 is the base. This is my attempt at a solution.. Log3(2x+3)-log3(x-2)=2 Log3(x-3/2)=2 Log3(x) - Log3(-3/2) = 2 Log3(x)-0.369=2 log3(x)=2.369 3^2.369=13.4 x=13.4 I plugged that into the original equation and I know it is not correct. Can anyone...

For the problem of differentiating ##y = x^5(3x-1)^3## using logarithmic differentiation, the solution provides the first step as rewriting the functions as ##\left |y \right | = \left | x \right |^5 \cdot \left | 3x-1 \right |^3##. This confuses me. First, how are we, mathematically, able to...

I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's definition of the natural logarithm function (page 234 -235) The relevant section of Stoll reads as follows: In this section we read: " ... ... To prove (a), consider the function $$L(ax), x \gt 0$$...

Hi I seem to have run into a strange problem. Suppose one wishes to maximize/minimize the function ## f(x) = (x-4)^{2} ##. Clearly, this function has a minimum at x = 4. One could find the extremum by taking the derivative and setting to zero. One could also compute the logarithm of this...

A theorem in my textbook states the following: For every n=0,±1, ±2, --- the formula ln z=Ln z ± 2nπi defines a function, which is analytic, except at 0 and on the negative real axis, and has the derivative (ln z)'=1/z. I don't understand why the logarithm isn't analytic for negative real...

Hello everyone, I am asked to calculate ##\log (e^{1+2i})##, and I would appreciate it if someone could verify my calculation.. My textbook defines ##\log z## as ##\log z = \ln |z| + i \arg z##. ##\log (e^{1+2i}) = \ln |e^{1+2i}| + i \arg(e^{1+2i}) \iff## ##\log (e^{1 + 2i}) = \ln|e e^{2i}|...

1) $( \log_{a}\left({b}\right)+\log_{b}\left({a}\right)+2) (\log_{a}\left({b}\right)-\log_{ab}\left({b)}\right)* \log_{b}\left({a}\right)-1=$?

$$(\sqrt{{\log_{n}\left({p}\right)} +\log_{p}\left({n}\right) +2} *(\log_{n}\left({p}\right)-\log_{np}\left({p}\right)) \sqrt{\log_{n}\left({p}\right)}$$

$$\log_{2}\left({24}\right) / \log_{96}\left({2}\right) - \log_{2}\left({192}\right) /\log_{12}\left({2}\right)$$

Solve the following $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$

hello why "you can’t take the logarithm of a negative number or of 0" ? thanks!

logarithm integral # (2) Solve the following $$\int^1_0 \frac{\log(1+x)\log(1-x)}{x}\,dx$$

Evaluate the following $$ \int^1_0 \frac{\log(1-x)\log^2(x)}{x}dx$$

For this function y=\sqrt{2ln(x)+1} if I use the chain rule properly, should I be getting this answer? \frac{dy}{dx}=\frac{2}{x} \times \frac{1}{2} \times \frac{1}{\sqrt{2ln(x)+1}} My aim of doing this is to verify that \frac{dy}{dx}=\frac{1}{xy}

Can you suggest a general analytical solution to the following equation \ln(x^{3/2})-bx-c=0 where x is real positive variable and b and c are real positive constants.

In...... it has beenshown in an elementary way that is ... $\displaystyle\int_{0}^{\infty}\frac{\ln x}{x^{2}+a^{2}}\ dx = \frac{\pi\ \ln a}{2\ a}\ (1)$The integral (1) isvery useful to shed light on the behavior of the logarithm functionin the complex field. Let's suppose we want to solve the...

Is the complex logarithm an example of a function analytic on the domain $1<|z|<2$ such that it's derivative on the domain is $\frac{1}{z}$? Thanks

Need to find derivative y = θ(sin(ln θ)) + cos(ln θ) My work θ(cos(ln θ))(1/θ) + sin(ln θ) + (-sin(ln θ)(1/θ)) (θcos(ln θ))/θ] + sin(ln θ) + ( (- sin(ln θ))/θ) cos(ln θ) + [θsin(ln θ) - sin(ln θ)]/ θ answer in book is 2cos(lnθ)

Hello. Can someone please check if I did this correctly. Question and my attempt are as the attached.

Hello, I'm solving the problems given in previous exams, and there's this question: Homework Statement a/ Give the value of ln(i), ln(-i) and i^i b/ If zo=-1-i , what is the value of lim [ ln(zo+e)-ln(zo+i*e) ] when e-> 0 Same question with zo=1+i Homework Equations The...

Given 3 constants a,b,c and 1 variable n $$\frac{log_a n}{log_b n} = C$$ , Prove it! I know that: $$log_a n = \frac{log_b n}{log_b a}$$ So for easier readability I let $$log_a n = \alpha$$ $$log_b n = \beta$$ $$log_b a = u$$ So here is what I got.. $$\alpha = \frac{\beta}{u}$$ <-- is my...

Problem: Given that $a,b$ and $c$ are the sides of $\Delta ABC$ such that $$z=\log_{2^a+2^{-a}} \left(4(ab+bc+ca)-(a+b+c)^2\right)$$ then $z$ has a real value if and only if A)a=b=2c B)3a=2b=c C)a-b=3c D)None of these Attempt: I am not sure where to start with this kind of problem. I wrote the...

I'm at one of those annoying stages where you know what the answer is, but you just can't seem to prove it... Would really appreciate some help with this one! The question is: What are the intervals where function f(x)=e2x-2ex is concave and convex respectively. I have derived f(x) to get...

In the context of complex number, how to prove that 1. $$log i +log(-1+i) \neq log i(-1+i)$$ 2. $$log i^2 =2log i$$

Given that $0.375<\log_x 5<0.376$ $0.256<\log_x 3<0.257$ $0.161<\log_x 2<0.162$ evaluate $\lfloor \log_x 7^{100} \rfloor$.

Homework Statement Hey everyone, So here's the problem, nice and simple. I have to find the following integral: \int_{0}^{\infty} \frac{x^{p}ln(x)}{x^{2}+1}dx, 0<p<1Homework Equations The only thing relevant is the residue theorem: \oint_{c}f(z)=2\pi i \times sum of residues enclosed The...

Homework Statement Hello, I have to calculate the derivative of y = log_x (x+1) so I used the formula of the derivative of a n-base logarithm and I get y' = 1/((x+1)logx) but that's wrong, why ? Thanks Homework Equations log_a x = 1/(xlog(a))

Homework Statement Hi, how can i show that the matrix logarithm log(I+A) is continuously differentiable on the set of matrices having operator norm less than 1. Homework Equations http://planetmath.org/matrixlogarithm The Attempt at a Solution i tried to compute the...

Hi, quick question. Is there a difference in notation when I say : (log8(x))^2 and log8(x)^2 Is it different or the same ? Thank you !

Homework Statement Solve for x 7^log(base: 7) 3 = x Homework Equations The Attempt at a Solution Ans is 3 Can someone explain to me how to solve for powers raised to a log like this one?

Homework Statement 6^log x=1/36 [b]2. Homework Equations Y=logcx The Attempt at a Solution How do u solve this? I know the 1/36 is the exponent. Usually the logs that u normally do is not in an exponent like this. The answer is 0.01. How did they get that?

I need to prove that $$H_n = \ln n + \gamma + \epsilon_n $$ Using that $$\lim_{n \to \infty} H_n - \ln n = \gamma $$ we conclude that $$\forall \, \epsilon > 0 \,\,\,\, \exists k \,\,\,\, $$ such that $$\,\,\, \forall k \geq n \,\,\, $$ the following holds $$|H_n - \ln n -\gamma | <...

Hello all, Am I right in assuming that log(\frac{3}{5}) = \frac{log3}{log5}? Thanks,

Log x ((x+3)/(x-1) > Log x x ?? I've managed to find 4 conditions for this inequality: 1. -1 > x > 3 2. x > -3 3. x > 0 4. x ≠ 1 but I'm not sure how to write the solution. Is it " 0 < x & 1 < 0 < 3 " ? Thanks.

I just need some help with some basic questions I can't remember from a long time ago, just started up school again... 1) Given a function f(x) = (quadratic on top)/(quadratic on bottom) When at x=1, I am given a tangent line to the function f(x), and also given the equation of the tangent...

I've been studying for a test and have been powering through the recommended problems and have stumbled upon a problem I just can't seem to figure out. $$\int_{0}^{\infty} \frac{logx}{1+x^{2}} dx$$ (Complex Variables, 2nd edition by Stephen D. Fisher; Exercise 17, Section 2.6; pg. 167)...