What is Logarithmic: Definition and 362 Discussions

A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a scale is nonlinear: the numbers 10 and 20, and 60 and 70, are not the same distance apart on a log scale. Rather, the numbers 10 and 100, and 60 and 600 are equally spaced. Thus moving a unit of distance along the scale means the number has been multiplied by 10 (or some other fixed factor). Often exponential growth curves are displayed on a log scale, otherwise they would increase too quickly to fit within a small graph. Another way to think about it is that the number of digits of the data grows at a constant rate. For example, the numbers 10, 100, 1000, and 10000 are equally spaced on a log scale, because their numbers of digits is going up by 1 each time: 2, 3, 4, and 5 digits. In this way, adding two digits multiplies the quantity measured on the log scale by a factor of 100.

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  1. U

    How Do You Solve This Logarithmic Integral Involving Cosine?

    Homework Statement \displaystyle \int_0^{2 \pi} x \ln \dfrac{3+ \cos x}{3- \cos x} dx Homework Equations The Attempt at a Solution Using property of definite integral 2I = \displaystyle \int_0^{2 \pi} 2 \pi \ln \dfrac{3+ \cos x}{3- \cos x} dx
  2. H

    Logarithmic Derivative of 2^log2(x^2+1) with Base 2

    Homework Statement 2^log2(x^2+1) 2 after the log function is actually the base. Homework Equations derivative of log(x)= 1/x derivative of a^x = a^x lna The Attempt at a Solution = 2^log2(x^2+1) ln2 *d[log2(x^2+1)]/dx = 2^log2(x^2+1) ln2 [2x/(x^2+1)ln2] ln2 in the...
  3. J

    MHB Integer ordered pairs in logarithmic equation

    no. of integer ordered pairs of $(x,y,z)$ in $ \sqrt{x^2-2x+6}\cdot\log_{3}(6-y) = x $ $ \sqrt{y^2-2y+6}\cdot\log_{3}(6-z) = y $ $ \sqrt{z^2-2z+6}\cdot\log_{3}(6-x) = z $ My approach :: Here $6-x,6-y,6-z>0$. So $x,y,z<6$ Now $\displaystyle \log_{3}(6-y) =...
  4. M

    Solving logarithmic and exponential equations

    Homework Statement I'm trying to solve trying logarithmic and exponential equations but for the life of me I can't seem to figure out what I'm doing wrong. Homework Equations Log equation: log3(x+2)+1 = log3x²+4x Exp equation: 2x + 2x+1 = 3 The Attempt at a Solution For the first...
  5. DreamWeaver

    MHB Logarithmic Integral on Stack Exchange - author unknown

    Here's an alternative solution to solving the integral:\int_0^1\frac{\log(1+x)\log(1-x)}{(1+x)}\,dx[Original post: calculus - A challenging logarithmic integral $\int_0^1 \frac{\log(1+x)\log(1-x)}{1+x}dx$ - Mathematics Stack Exchange] I'll post a link on there so the OP can see my work...
  6. DreamWeaver

    MHB Logarithmic Integrals, Polylogarithms, and associated functions

    This is not so much a tutorial, but rather a collection of useful results and techniques. Some of the proofs will be quite long, since as much as possible, I'll aim to prove most results and functional relations used herein, rather than just present another's identity as fact. There will be a...
  7. A

    Logarithmic Ratios: Solving Input/Output & dB to Np

    The current input of a system is 250mA. If the current ratio of the system is 15dB, determine: (i) the current output and (ii) the current ratio expressed in nepers I have the answer to the questions which are (i) 1.406 A (ii) 1.7272 Np I am just struggling with the complexity of the...
  8. N

    MHB Quadratic and logarithmic models (new post)

    Here is my previous post! please look at pages 3 - 4 to see what I am talking about. http://mathhelpboards.com/pre-algebra-algebra-2/mathematical-modeling-6006-4.html I figured out how to input the quadratic and logarithmic equations into wolfram, this is what i got. Let me know if its correct...
  9. N

    MHB Are my answers correct? Quadratic and logarithmic modeling

    This is what i got... For option 1 I think it's y=( .5(x-20))^2+20 for x>20 For option 2 I think it's y=30+.5*ln|x-25| for x>25Question: develop equations that model the two replacement pack options (option 1) quedratic model, monthly access fee \$20, included gigabytes 20, cost per...
  10. M

    Tryimg to understand this logarithmic identity

    (I know the title contains a typo, but I can't edit it!) I'm trying to understand and/or prove this identity: blogbx = x I've inserted numbers, and it does work, but I just don't seem to understand why. I mean, some identities are obvious, like: logb bx = x since bx = bx but I can't make...
  11. G

    Logarithmic Question: Solving for Unknowns Using Logarithm Properties"

    Homework Statement Hi, it maybe stupid question, but I struggle on this problem.. log16(4354)+3/log25(43)=a+b*log25 Homework Equations Using properties: logabn=n logab logn(bc) = lognb+lognc logba=1/logab The Attempt at a Solution I can't find the b number. log16(4354)+3/log25(43)=6...
  12. MarkFL

    MHB Tangent Line & Log Diff EQs: Jawairia's Qs at Yahoo Answers

    Here are the questions: I have posted a link there to this topic so the OP can see my work.
  13. alyafey22

    MHB Fractional Logarithmic Integral 02

    \int^1_0 \frac{\log (1+x)}{1+x^2} dx $\log $ is the natural logarithm .
  14. G

    Are all divergences in quantum field theory logarithmic?

    In Zee's QFT book he writes an amplitude as: M(p)=\lambda_0+\Gamma(\Lambda,p,\lambda_0) He then states that you make a measurement: M(\mu)=\lambda_0+\Gamma(\Lambda,\mu,\lambda_0) \equiv \lambda_R and substitute that into M(p) to get...
  15. C

    Self energy logarithmic divergence due to chiral symmetry.

    In peskin at page 319 right above equation (10.6) he writes "If the constant term in a taylor expansion of the self energy were proportional to the cutoff ##\Lambda##, the electron mass shift would also have a term proportional to ##\Lambda##. But the electron mass shift must actually be...
  16. C

    MHB Proving logarithmic identity

    Hello MHB. How can i proof this equation? log(x).log(x^2).log(x^3)... log(x^90)=4095
  17. M

    Do not get Logarithmic antiderivatives

    Dear All, the antiderivative of $$\frac{1}{x}$$ is $$ln \vert x \vert $$. If I then consider the expression $$ (x-a) ln (\vert x-a\vert)-x $$ and compute the first derivative I obtain $$ ln (\vert x-a \vert)$$. If I then consider the equivalent expression $$ (x-a) ln (\vert a -x\vert)-x $$...
  18. D

    Derivatives of General Exponential and Logarithmic Functions

    So in my math class we're studying derivatives involving ln(), tanh, coth, etc.. I need to say this first. I skipped precalc and trig and went straight to calculus, so whenever I see a trig problem, I can only go off of what I've learned "along the way." This problem has baffled me, please...
  19. H

    MHB Finding x in Logarithmic Equation

    Hi, sorry if it's not in the right subforum. idk how to solve x: http://puu.sh/2Lbb1.png The answer is x = -1.70951. how do we get there? please explain everystep. thanks :3 ****someone made it this far, idk if it is the correct path: log2 (2^(x-1)+3^(x+1)) = 2x - log2 (3^x) log2...
  20. C

    Origin of Logarithmic Functions

    Ive repeatedly asked people and my teachers how to solve logs by hand, and I've always got the same answer "They used tables and/or taylor series" but when i ask how the tables were made no one seems to know. So i am curious as to how John Napier and other mathematicians at the time found the...
  21. E

    Solving Logarithmic Spirals for Length from Origin to Point X

    Homework Statement This is a logarithmic spiral. How can I find the length of the spiral (curve) from the origin to point X on the curve? The distance between the origin and the point is given (we can call that the radius) and the distance between the origin and point A is given which this...
  22. M

    Derivative using Logarithmic Differentiation

    Homework Statement http://www.flickr.com/photos/50468243@N02/8577233960/ Homework Equations The Attempt at a Solution In the pic, the noticeable difference between my answer and Wolframalpha's is the tan[log(sin x+1) to base 5]. If I do logarithmic differentiation, what I get is...
  23. S

    Solve Logarithmic Equation: 3^(2x) - 5(3^x) = -6

    Homework Statement 3^(2x) - 5(3^x) = -6 Homework Equations The Attempt at a Solution I can't determine the right answer :/? I'm not sure what's incorrect with my current method. 3^(2x) - 5(3^x) = -6 log((3^(2x)) - log((5(3^x)) = -log(6) xlog(9) - xlog(15)= -log(6) x((log(9) - log(15)) =...
  24. marcus

    Riello: EPRL radiative corrections only logarithmic in cosmo constant

    http://arxiv.org/abs/1302.1781 Spinfoam transition amplitudes only depend lightly on cosmological constant. This is looking good (especially in view of the Hamber Toriumi paper that just appeared.) ==quote Aldo Riello's new paper, page 4== The ERPL-FK model can also be extended to the case of...
  25. T

    MHB Struggling with Logarithmic Expansion - Can You Help?

    Hello guys, I am finding difficulties expanding log[(x+epsilon)^2 + y^2] as epsilon->0. Could anyone help me with the expansion? Kindest regards
  26. N

    Equation with logarithmic and polynomial terms

    This is not actually a homework question, but it seemed appropriate to put it here. In an old exam from 1921 I found the following problem. I never learned how to solve this type of thing and I haven't been able to figure it out, so: how does one solve this? Homework Statement Solve for...
  27. M

    Problem with a logarithmic rule

    Please could some one help me with the rules of logarithms to understand why inverting the numbers in the bracket and change the minus sign to a plus sign gives the same result? Thank you
  28. I

    Mathematica Solving a 2D Parametric Plot of Logarithmic Spiral

    I am stuck on the following task; Create a 2D Parametric Plot showing a spiral path. The parametric equations for a logarithmic spiral are x=k^u Cosine(u), y=k^u Sine(u), where k is a constant, and u is the plot parameter. What does the value of k determine? I have been typing in the...
  29. D

    Derivative of Exponential and Logarithmic Functions

    Homework Statement Find the equation of the tangent line to the graph of y = lnx2 at the point (2, ln4) Homework Equations The Attempt at a Solution y' = (lnx2)' y' = 2lnx y' = (2)(lnx)' + (lnx)(2)' y' = 2/x Mtan/(2,ln4) = 2/2 = 1 Equation of tan line: y - ln4 = 1(x-2) y = x - 2 + ln4
  30. D

    Derivative of Exponential and Logarithmic Functions

    Homework Statement PROBLEM 1 Find slope of tangent line at (1,0) g(x) = x2lnx Solution: g'(x) = (x2lnx)' g'(x) = x2[1/x . (x)'] g'(x) = x2 . 1/x g'(x) = x M tan/(1,0) = 0 PROBLEM 2 Use implicit differentiation: yex+ y2 = 7lny - x3 Solution: (ex)(y') + 2y(y') - 7(y'/y) = -ye - 3x2...
  31. anemone

    MHB How to prove this logarithmic inequality?

    Hi all, I've been having a hard time trying to solve the following inequality: Prove that $\displaystyle \left(\log_{24}(48) \right)^2+\displaystyle \left(\log_{12}(54) \right)^2 >4$ I've tried to change the bases to base-10 log and relating all the figures (12, 24, 48, and 54) in terms of 2...
  32. M

    Use logarithmic differentiation to find the derivative

    Hi .. Use logarithmic differentiation to find the derivative can please check my answer and How I can know if the question want answer by using logarithmic differentiation or not ?
  33. A

    MHB How are Earthquake Magnitudes Calculated on the Richter Scale?

    The magnitude of an earthquake, measured on the Richter scale, is given by R(I)= log(I/I0) Where I is the amplitude registered on a seismograph located 100km from the epicenter of the earthquake, and I0 is the amplitude of a certain small size earthquake. Find the Richter scale ratings of...
  34. A

    MHB How to Solve Logarithmic Equation log4x-log4(x+3)=-1?

    log4x-log4(x+3)=-1 This is what I have so far: =log4(x)(x-3)=-1 =(x)(x-3)= 4-1 =(x)(x-3)= 1/4 = x2+3x-1/4=0 Now do I use the quadratic equation to solve? Thanks.
  35. C

    Confusion on this logarithmic equation NEED HELP

    e^(3+2logx) = (3x-2)*e^3 I can get down to this point: e^(logx^2)=3x-2 I have checked the solution in the book, and the steps. Turns out it simplifies to x^2 -3x +2=0. The problem is I can't figure out how I pass from e^(logx^2) to x^2, because as far as I know logx isn't the same as lnx...
  36. G

    Does logarithmic decrement depend on initial amplitude?

    Lets say we have a weight attached to a spring. When releasing it under water (whole motion occours under water) at different initial amplitudes, will logarithmic decrement be the same? And will the period change? I think it should be the same for different initial amplitudes. But...
  37. U

    Why Do I Get a Different Result When Differentiating ln(x + sqrt(x^2-1))?

    How do I find the derivative of (x)=ln(x+(x^2-1)1/2) The answer is suppose to be 1/(x2-1). But I keep ending up with 2x/(x2-1).
  38. V

    Logarithmic Differentiation of (x+5)(x+9): Where Did I Go Wrong?

    Homework Statement Find the derivative using logarithmic differentiation: y=(x+5)(x+9) The Attempt at a Solution lny=ln(x^2+14x+45) lny=(2x+14)/(x^2+14x+45) y'=(x^2+14x+45)((2x+14)/(x^2+14x+45))However, I know the derivative of the function is actually 2x+14. So I am wondering what is wrong...
  39. Z

    How to Solve Complex Logarithmic Equations

    Homework Statement A) Solve LogX^(LogX) = 4 B) Log3 X - Log27 X = 4/3Homework Equations Basic 3 log rules: 1. Logc(MN) = LogcM + logcN 2. Logc(M/N) = LogcM - LogcN 3. LogcM^p = pLogcM The Attempt at a Solution I have no idea how to start either.
  40. T

    Solving a logarithmic equation.

    Homework Statement log3x+log3(2x+1)=1 Solve for x. Homework Equations The Attempt at a Solution log3((2x+1)x)=1 3log3((2x+1))=31 (2x+1)x=3 2x2+x=3 2x2+x-3=0 Using the quadratic formula x= (-1±√(12-4(2)(-3)))/(2(2)) = (-1±√(1+24))/(4) = (-1±√25)/(4) = (-1±5)/(4)...
  41. D

    The derivative using logarithmic differentiation. Is this correct?

    1. Homework Statement [/b] Find the derivative of the given function. Homework Equations Chain rule and logarithmic differentiation. The Attempt at a Solution See attached .gif. I was just wondering if this seemed correct? Thanks!
  42. 9

    Derivatives of logarithmic -

    Derivatives of logarithmic functions - please help Homework Statement I am having trouble differentiating logarithmic functions. I am following this basic rule they gave us, namely: if y = ln g(x) then y' = g'(x)/g(x), but it is not working :(. Where am I going wrong? Homework Equations...
  43. C

    MHB Equation Logarithmic: Solve x | MHB Help

    Hey MHB i tried but i can't find the solution for this equation, please help me. {27x}^{\log_5 x}={x}^{4}
  44. G

    MHB Solving for a in Logarithmic Equation: x(x-15)=2

    In log x + (x-15)=2 and I have to convert it to an exponental. On the practice one they said in the form of log(small)M+log(small)aN=log(small)aM*N so log(small)ax a^x=x a=10 Why in this equation does a=10?? I don't understand. In the equation above I understand it being x(x-15)=2 just not the...
  45. S

    Parametric Equation for Tangent of Logarithmic Spiral

    [a]Give a parametric equation for the line tangent to this curve at t = \frac{pi}{4}. \vec{r(t)} <e^tcost, e^tsint> Give the equation for this same tangent line in the form ax + by = c [b]My attempt \vec{r(\frac{pi}{4})} = <e^\frac{pi}{4}cos\frac{pi}{4}, e^\frac{pi}{4}sin\frac{pi}{4} =...
  46. C

    Logarithmic Scale: What Happens When Plotting x,y Values?

    Hello people, I have a question about the log-scale. What happens when we switch a plot from linear scale to the log scale? Let's say I have two arrays: x values and corresponding y values. I plot them using a linear scale and then I switch to the log scale. What happens? Does the program...
  47. J

    Logarithmic Decay of Hypochlorite/Chlorine by UV light

    Hi, Does anyone know why the decay of hypochlorite/free chlorine in pool water due to UV light might show a logarithmic decay (natural log) as opposed to an exponential one? I did a chemistry experiment with scaled up concentrations of chlorine and cyanuric acid (scaled up 25x). Even in...
  48. M

    Solving a Logarithmic Spiral Puzzle - Martine

    Homework Statement a logarithmic spiral is given. The center lies on the x-axis. after a turn of 180 degrees counterclockwise I am 86.23m away from the starting point, after 360 degrees I'm 75.41m away from the start (radius, not along the spiral). Where am I after I walked exactly 3km along...
  49. S

    Is This Example Correctly Solved Using Logarithmic Differentiation?

    Something about this worked problem looks off. Is this example correctly solved using logarithmic differentiation? The original problem is y = (2-x)^(sqrt x). If anyone who is rather confident with this could double check this example it would really help me out. Thanks. I attached the...
  50. C

    Help with Logarithmic Differentiation

    Homework Statement y = [2x + 1]^5 * [(x^4) - 3]^6Homework Equations I take the derivative of the natural log of both sides: (y' / y) = [(10 ln(2x + 1)^4) / (2x + 1)] + [(24x^3 ln(x^4 - 3)^5) / (x^4 - 3)] then I multiply both sides by the original function: y' = [((10 ln(2x + 1)^4) / (2x +...
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