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. T

    Median interpolation on logarithmic data

    Homework Statement Hello, I have a set of data that is grouped into bins. The bin sizes increase logarithmically by 10% (1.1 x the previous bin). Homework Equations Bin Mean Radius Frequency 1.03 4924 1.10 9938 1.21 14009 1.32 12269 1.44 15813 1.58 18723 1.74 21471 ...
  2. K

    Integral of 1/lnx: Calculate Here

    what is the integral of the function 1/lnx?
  3. Z

    Would a theory with ONLY logarithmic divergencies be renormalizable ?

    the idea is let us supppose we have a well behaved theory where only logarithmic divergences of the form \int_{0}^{\infty} \frac{dx}{x+a}=I(a) for several values of 'a' occur , then would this theory be renormalizable ?? , i think QED works because only logarithmic divergencies appear ...
  4. T

    Torsional pendulum, logarithmic decrement

    Homework Statement The amplitude of a torsional vibration decreases so that the amplitude on the 100th cycle is 13% of the the amplitude of the first cycle. Determine the level of damping in terms of the logarithmic decrement. Homework Equations The Attempt at a Solution Is this...
  5. T

    Torsional Pendulum, Logarithmic Decrement

    The amplitude of a torsional vibration decreases so that the amplitude on the 100th cycle is 13% of the the amplitude of the first cycle. Determine the level of damping in terms of the logarithmic decrement. Is this simply ln(100/13)= 2.04 or ln(13/100)= -2.04?
  6. J

    Solve Logarithmic Equation: log9 (p) = log12 (q) = log16 (p+q)

    log9 (p) = log12 (q) = log16 (p+q) What is q/p? 9^x + 12^x = 16^x calculate x (4/3)^x = q/p I guess that would work if I'd know how to solve that equation (without using a calculator), unfortunately I don't... 3^(2x)+4^x*3^x = 4^2x
  7. F

    Logarithmic Galactic Potential

    Hey guys, I have a potentially (no pun intended...ok maybe a little) stupid question. People often use log potentials to model galaxies with flat rotation curves and things and I'm currently using it to model a galactic potential for some simulation work. However none of the sources i have...
  8. F

    Finding N: Logarithmic Approach Needed

    Homework Statement I need to find N: {[pi^(N+1)] / (N+1)!} . {1/2^N} <= 10^-5 Homework Equations The Attempt at a Solution I'm really struggling with this, i think taking logs is involved, but i can't seem to find a value of N. Any help?
  9. R

    Expressing a logarithmic equation

    Homework Statement http://img142.imageshack.us/img142/8564/76203773.jpg Homework Equations I need to somehow express the above equation in terms of logax, logay,and logaz.The Attempt at a Solution I got the answer loga * [ |x|*sqrt (x)] + loga * [|y|] - loga * [z] But I'd just like to get...
  10. T

    Find the exact logarithmic form of y at A where y= -x + tanh4x is

    \frac{2\sqrt{3} - ln(2+\sqrt{3})}{4} Given that the x co-ordinate is ln(2+ \sqrt{3}) which I worked out in the first part of this qusetion given that at A there is a maximum turning point (so maximum turning point means dy/dx = 0), but I'm not sure that helps in this part of the question y=...
  11. S

    Calculating Radioactive Decay: Finding the Time for 80% Decay

    Homework Statement The half-life of a radioactive substance is 194 days. How many days will it take for 80% of the substance to decay? Homework Equations The Attempt at a Solution I had to make 1/2n=1/5. I did this by finding log25=n. After this, I needed to only multiply by...
  12. S

    Just a quicky, which one of these two is correct? Logarithmic integrals.

    Homework Statement \int\frac{ds}{s^2} Homework Equations None really...I suppose \int\frac{a}{bx + c} = \frac{1}{b}ln.a.(bx + c) The Attempt at a Solution Pretty much, using the formula above gives you: ln(s$^{2}$) But I was thinking, if you rearrange \frac{ds}{s$^2$}...
  13. K

    Solve Log Equation: Find Value of x

    Homework Statement Find the value of x Homework Equations log x - 1/2log(x-1/2) = log(x+1/2) - 1/2log(x+1/8) The Attempt at a Solution used the log a + log b = log a*b & log a - log b = loga/b and got a cubic equation 4x^3-8x^2-4x-1 = 0 but the answer in the book is x=1
  14. X

    Solving a logarithmic equation

    Homework Statement (e^x)^2-5(e^x)=0 Homework Equations I've reviewed my logarithmic rules, but I cannot get to the right solution. I should be trying to get one e^x on each side so that I can take the ln of both sides and end up with a simple algebraic process to solve. The Attempt...
  15. V

    Intense Logarithmic Diferentiation Question

    Homework Statement Hey guys, this is my first post and of course first question to ask of calculus Alright well I had this test and we had this very difficult question that I could not solve, it was in the hardest section. The question is as follows: A projectile thrown...
  16. J

    Logarithmic Scale: Calculating Evenly Spaced X Values

    Hello All, I am working on a program that produces plots that use a logarithmic scale on the X-Axis (for showing decade frequency ranges). The Y-axis can have any arbitrary value and I'm not thinking about that at this point, but here is my question. Say, for simplicity that every Y-value...
  17. R

    Solving for \Pi: Unraveling the Logarithmic Equation

    Hi, I have the following equation: \Theta = log \left(\frac{\Pi}{1-\Pi}\right) I want to re-arrange it for \Pi Here's my attempt: \Theta = log \left( \frac{\Pi}{1-\Pi}\right) \Theta = log \left( \Pi \right) - log \left(1-\Pi \right) exp^{\Theta} = \Pi - (1-\Pi)...
  18. T

    Solving Logarithmic Equation: Why Isn't It Working?

    Homework Statement I can't seem to get my head round this problem, I know how to use the 'change the base rule' log_2 x + log_4x = 2 \frac{logx}{log2} + \frac{logx}{log4} = 2 why is this not correct ??
  19. K

    Solving cosz=2: A Logarithmic Approach

    Homework Statement Sove cosz=2 Homework Equations The Attempt at a Solution I need to solve this, but am not quite sure how. I was told to use log.
  20. J

    Logarithmic Differentiation Help NeededA LOT OF PROCESS WORK IS SHOWN

    Use logarithmic differentiation to find: a.) d/dx of [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))] b.) d^2/dx^2 (sech^-1(e^(2*x))) work shown for a: let y= [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))] taking the natural logarithm of both sides: ln y= ln...
  21. J

    Logarithmic differentiation help needed work shown

    Use logarithmic differentiation to find: a.) d/dx of [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))] b.) d^2/dx^2 (sech^-1(e^(2*x))) work shown for a: let y= [(sin^-1(x^2)*sinh^-1(x^2))/(sin^4(x^2))] taking the natural logarithm of both sides: ln y= ln...
  22. M

    Where can I find a complete logarithmic formulas worksheet or tutorial?

    Homework Statement I need to find as much as a complete logarithmic formulas worksheet or tutorials I have not been in school for atleast 2 semesters and the class I am taking right now makes use of the basic logarithmic functions, not to complicated from what the teacher has told us. I do...
  23. S

    Logarithmic Differentiation

    y = (sin2x)3(x4-4x)6 divided by (cosx) + e3x i came out with an answer y' = (sin2x)3(x4-4x)6divided by (cosx) + e3x [3cot2x + 24x3-24 divided by x4-4x + Tanx + 3x could someone tell me if I am right? i don't know if this is what I am supose to do, if you want me to write out...
  24. C

    Another logarithmic diff. problem and graphing question

    Homework Statement Find a formula for f^(n)(x) if f(x) = ln(x-1) Homework Equations The Attempt at a Solution Not completely sure what the problem is asking for, any suggestions? Also, another quick question. If I am trying to graph a problem using a ti-89 that shows two tangent...
  25. C

    Help on logarithmic differentiation problem

    Homework Statement Use logarithmic differentiation to find the derivative of the function. Homework Equations y = (sin(x))^(ln(x)) The Attempt at a Solution I guessing the first step is raise both sides to "e", but so far I have only completed problems by taking the natural log...
  26. N

    Resolution of a logarithmic function

    Homework Statement Resolve the following equation: log2 x + log2 [(x-3)/5)] = 3 Homework Equations Logarithmic laws: logc m + logc n = logc mn The Attempt at a Solution Restriction: (x-3)/5 > 0 x-3 > 0 x > 3 Resolution: log2 x + log2 [(x-3)/5)] = 3 log2 x[(x-3)/5)] = 3 log2 [(x2 - 3x)/5x]...
  27. N

    What is the simplified form of 2 log (x2 - 1) - log (x + 1) - 2 log (x - 1)?

    Hello again.. Homework Statement I I have to change this to one term only, the valid answer is: log (x + 1) The term is: 2 log (x2 - 1) - log (x + 1) - 2 log (x - 1) Homework Equations Logarithmic laws: x log n = log nx log m - log n = log (m/n) The Attempt at a Solution I...
  28. N

    Solve Logarithmic Equation: [log_{2} 9/log_{2} 10] - log 3

    Homework Statement I have to take the following equation and make it one single log [log_{2} 9/log_{2} 10] - log 3 The answer is equal to: log 3 I usually know how to do this however I have no clue in this case, I have the answer however. Homework Equations None? log x - log y...
  29. J

    Help Needed: Solving Logarithmic Equation

    hello.. i don't know where to start solving this problem can you please help me? log phi of x (v)= e-2vx^(1+v/sq.rt.x)2x-1. please..I really need your help. Thank you so much.
  30. N

    Solve Logarithmic Equation: 4(x+1)= 8(x-1)

    Just wondering if someone can tell me if I got this correct. It was a problem on my midterm a month ago and I am going over the probs I got wrong, and I don't know the correct answers. Homework Statement 4(x+1)= 8(x-1) Homework Equations The Attempt at a Solution log 4(x+1) =...
  31. N

    Logarithmic Equation with x on both sides

    Exponential Equation with x on both sides Homework Statement 18x^2=6e^(2x) Homework Equations I don't know any for the step I got stuck on. The Attempt at a Solution 3x^2=e^2x ln3x^2=lne^2x 2ln3x=2x ln3x=x
  32. jacksonpeeble

    Exponential and Logarithmic Equations and Modeling Problems

    Hello everyone! I have two precalculus problems that are from an assessment today that I had some great trouble with. I wrote the objectives below, followed by the problems themselves. Any help is greatly appreciated! Homework Statement 1. Solve exponential and logarithmic equations when...
  33. J

    Trouble with Logarithmic Equations: Help Needed for 5 Questions

    I'm having a lot of trouble on this worksheet I have, I've got most of the 32 questions except for about 5. I won't type out all the laws of logarithms as I assume that anyone coming in here to help me already knows them. So here are my questions (By the way, I've done work for them I just...
  34. P

    Solve simple logarithmic question

    Homework Statement Ae^dt + Be^ft + Ce^gt + De^ht + ... = Y Where A-Y and d,f,g,h, etc are constants. Homework Equations Logarithmic identities... ( log(AB) = log(A) + log(B), log(A^x) = x log A, etc) The Attempt at a Solution I could do this if there was only one term on...
  35. L

    Finding the Derivative of a Logarithmic Function with a Variable Exponent

    Another problem I'm not sure of :( find \frac{dy}{dx} for the function xy^{2} + x lnx = 4y my answer y^{2} + x2y \frac{dy}{dx} + lnx + x (1/x) \frac{dy}{dx} = 4\frac{dy}{dx} x2y \frac{dy}{dx} + \frac{dy}{dx} - 4\frac{dy}{dx} = -y ^{2} - lnx \frac{dy}{dx} ( x2y - 3) =...
  36. F

    Derivatives of natural logarithmic functions

    f(x) = ln (12x-5/9x-2) f'(x) = (4/3) (1/ln10)(9x-2/12x-5) Is this correct??
  37. F

    What is the logarithmic equation for 3e3y-6 = 2x2-1?

    3e3y-6 = 2x2-1 ln3e3y-6=ln2x2-lne ln3e3y-6=ln2x2/lne 3e3y-6=2x2/e 3e3y-6*e=2x2 e3y-6=(2/3)x2 3y-6=ln((2/3)x2) 3y=ln((2/3)x2)+6 y=(1/3)ln((2/3)x2)+2 I wonder the answer is correct, the question is asking about the equation of y.
  38. R

    Solving Logarithmic Series Homework: Find x Values

    Homework Statement x=\frac{1+\frac{f(x+1)}{1!}+\frac{f^{2}(x+1)}{2!}+\frac{f^{3}(x+1)}{3!}+...}{1+\frac{f(x)}{1!}+\frac{f^{2}(x)}{2!}+\frac{f^{3}(x)}{3!}+...} f(x) is a twice differentaible equation. 1. Find the possible values of x when; f(x+0.5)<f(x-0.5) 2. Find the possible values of...
  39. R

    Minimum value of logarithmic equation

    Homework Statement For x>1, find the minimum possible value of 2 log_{2}x-log_{x}(0.01) The Attempt at a Solution Greater the value of x, greater is the value of expression. Right? I tried to differentiate it, but it was no help. The derivative becomes zero when |log x|=\sqrt{log 2}...
  40. B

    Derivative with general exponential and logarithmic fuctions

    Homework Statement f(x)=(((x^7)+3)((x^3)+2)) / (((x+3)^2)((x^6)+5)) Find f'(x) Homework Equations d/dx lnx = 1/x d/dx ln(f(x)) = f'(x)/f(x) The Attempt at a Solution ln(f(x)) = ln((x^7)+3) + ln((x^3)+2) - ln((x+3)^2) - ln ((x^6)+5) f'(x)/f(x) = (7x^6)/((x^7)+3) +...
  41. B

    Minimum of a logarithmic function

    Homework Statement Find the maximum and minimum values of the function F(x)=ln(x/(x^2+1)) on the interval (0,10] Homework Equations d/dx[lnu]=u'/u The Attempt at a Solution f'(x)=(1-x^2)/[(1+x^2)x] f'(x)=0 when 1-x^2=0 1=x^2 x=1 which is the maximum f(1)=-.693 what do I use...
  42. R

    Logarithmic differentiation

    Question: Find derivative of f(x)=((x^2)(x^3))/((x^4)(x^2)) Attempt: ln f(x)=(lnx^2)+(lnx^3)-(lnx^4)-(lnx^2) Can someone tell me what I have done wrong so far? Thanks
  43. A

    How would you prove that constant is a subset of logarithmic?

    O(1) is constant O(log n to the base 2) is logarithmic O(n) linear how would you prove that constant is a subset of logartihmic?
  44. R

    Solving Natural Logarithmic Equations

    Homework Statement Solve ln(2x+1)=2-ln(x) for x. Homework Equations The Attempt at a Solution e^(ln(2x+1)) = e^((2-ln(x)) 2x + 1 = e^(2/x) 2x^2 + x = e^2 2x^2+x-e^2 = 0 At this point, I know you're supposed to use the quadratic equation. But, my problem is with how...
  45. A

    Exploring Exponential and Power Law Variations with Logarithmic Plots

    [SOLVED] Logarithmic plots... Homework Statement use logarithmic plots to test exponential and power law variations This statement appears in the Cambridge A'Level Syllabus Can somebody please explain what does this statement require from the student? Helpful links would be highly...
  46. F

    Understanding Logarithmic Function Problems: Solving for x

    Homework Statement solve x. Log(35-x^3)/Log(5-x)=3 Homework Equations Log(a/b)=Log a - Log b The Attempt at a Solution The answers would be 3 or 2 (so I'm guessing it has sumthing to do with quadratics) i tried to expand the left side first, then rearrange it to Log(7/x^4)=3...
  47. B

    How to Evaluate the Derivative of a Logarithmic Function with a Base of 10?

    Homework Statement if g(t)=(10^t)(log...t) then evaluate g'(10) ......10 <---------(my attempt at a log base 10) Homework Equations im completely lost...i don't know if i should take the ln of both sides...or what to do really. The Attempt at a Solution
  48. R

    What is the Antiderivative of x^(-2) with Given Conditions?

    Find f. (x>0) f''(x) = x^(-2) x > 0 f(1) = 0 f(8) = 0 Alright, everything was fine until one point...okay. First derivative: f''(x)=x^{-2} f'(x)=-x^{-1}+C Now here's where I'm not sure: in the next step, it will be x raised to zero, which is one. times 1/0, which is zero...
  49. N

    Investigating Logarithmic Singularity in 2D Tight Binding Approximation

    Homework Statement Given some dispersion relation for the tight binding approximation in 2D: e(k_x,k_y) = -2t_1[cos(k_x*a)+cos(k_y*a)]-4t_2[cos(k_x*a)cos(k_y*a)] Show that the density of states has a logarithmic singularity for some choice of parameters t_i. Homework Equations g(e)de=g...
  50. A

    What is the derivative of ln(x^2 + y^2)?

    Find y' if y= ln( x^2 + y^2) I thought this was just a regular natural log derivative combined with the chain rule. So what I got was (2x + 2y)/ (x^2 + y^2) But this wasn't the correct answer. So could someone help point out my mistake. Thank you.
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