Logarithmic Definition and 362 Threads

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. S

    Summation of a Logarithmic Series

    Homework Statement What is the sum of the following series? log(3/2)+log(4/3)+log(5/4)+...log(200/199). Where log(x) is log base 10 of x. Homework Equations The Attempt at a Solution Evidently, the previous form equals: log(3/2*4/3*5/4*...200/199) I'm missing something -...
  2. M

    Logarithmic and exponential integrals

    In this text, I will ask a question about the power series expansion of exponential and logarithmic integrals. Now, to avoid confusion, I will first give the definitions of the two: \mathrm{Ei}(x)=\int_{-\infty}^{x}\frac{e^t}{t}dt \mathrm{Li}(x)=\int_{0}^{x}\frac{dt}{\log(t)} where Ei denotes...
  3. B

    Problem of Calculus: derivative, i guess logarithmic differentiation

    Homework Statement Image of the problem: http://prntscr.com/addkf Homework Equations My question is how I can solve the equation I gave above. Should I use logarithmic differentiation? Because I think that the logarithmic differentiation is used when y = (the equation) but my problem is f...
  4. A

    Exploring the Relationship Between Inverse Variables on a Logarithmic Scale

    Let me start by saying my understanding of logarithms is relatively limited, with that out of the way on to my question. The question I have is in a situation where you have 2 items that are inversely related and both have a starting point of 100%. How would you calculate the point where...
  5. T

    How to draw a logarithmic spiral

    I have a simple geometry question hopefully I'm posting this in the right forum. I want to draw a simple logarithmic spiral using only compass and ruler or any other simple mechanical means. The requirement is that each spiral must be halfway from the center to the next spiral. Is there a...
  6. T

    Master Logarithmic Functions: Solve Equations with Integers | 3(6x-1) = 1 + 4/6x

    Homework Statement Solve 3(6x-1) = 1 + 4/6x giving your answer in the form x = a - log6b where a,b are integers (Z) Homework Equations The Attempt at a Solution 3(6x)(6-1)(6x) = 6x + 4 Let 6x be y, (1/2)y2 - y - 4 = 0 y = 1 ± √(1+8) = 1 ± 3 3x = 4 xlog63 = log64 What...
  7. M

    Dark energy from logarithmic corrections to cosmological horizon entropy?

    We currently have a thread about logarithmic corrections to the basic black hole entropy formula. I was thinking about attempts to relate the magnitude of dark energy to the area of the cosmological horizon, and about the various analogies made between the cosmological horizon and the horizons...
  8. M

    Logarithmic corrections to black hole entropy, especially for the standard model

    Ashoke Sen has written a paper http://arxiv.org/abs/1205.0971 which uses path integrals in Euclidean gravity to compute "Logarithmic Corrections to Schwarzschild and Other Non-extremal Black Hole Entropy in Different Dimensions". Sen starts by listing several varieties of extremal black...
  9. B

    What is the measurement of a Logarithmic

    I was reading a book and it said the sun was 10^12 Logarithmic. What is the measurement of a Logarithmic?
  10. L

    Exponential and Logarithmic Functions

    Can anyone check my work? I'm doubtful of my answer. Homework Statement The bacteria in a 4-liter container double every minute. After 60 minutes the container is full. How long did it take to fill half the container? Homework Equations I used: F = A × 260 (1/2)F = A × 2x F...
  11. B

    Calculating the gradient of a logarithmic scale

    Homework Statement hi i have the folowing data i would like to plot in matlab plotERRLW = 0.0466 0.0111 0.0074 0.0046 NX = 50 500 1000 2000 i am using a logarithmic graph to gain a straight line, if i wished to find the gradient of the line...
  12. A

    Exponential Growth and Logarithmic Growth

    I'm a very conceptual person, and I've been reading about exponential and logarithmic growth but don't fully have the kind of conceptual grasp on the two and how they differ that I'd like, so I"m curious: What is the difference between exponential growth and logarithmic growth and what causes...
  13. D

    Solving 4log45 Logarithmic Problem

    Homework Statement 4log45 Homework Equations ay = x ~~~> loga(x) = y The Attempt at a Solution I've separated the problem and decided to solve the logarithmic section first giving me: 4x = 5 I could move forward if I wasn't stumped on how to figure out what x is in this...
  14. M

    Help with this logarithmic equation

    2n *logn =0.1 *n^2 solve for n please and thanks! pozhalste e spaciba! por favor a obrigadao
  15. F

    Logarithmic Integral and Primes

    Hey guys, I was reading a brief article which described the logarithmic integral for approximating π(x) in two ways: ∫x/log(x) dx and ∫1/log(x) dx I am aware that the second is the actual definition of li(x) but the top is used extremely frequently and upon trying out the top it...
  16. A

    Formula to Balance Logarithmic or Continuously Compounding ?

    If I was interested in determining how to balance the rate of reinvestment of profits back into the original investment (to add to its compounding potential) while still growing the profit return on the investment. How would I go about calculating something like that ?
  17. D

    Need help understanding logarithmic properties

    Okay, so in class I learned that: 10^log(4x) = 4x But I don't understand why. I get that a log without a subscript is considered base 10, so: 10^y = 4x Is the way to understand "log(4x)", right? What if the problem was a different base? Would the "10" coefficient and 'log' in...
  18. B

    Master Logarithmic Equations: Expert Tips for Solving Problems

    Having problems with these questions, what's putting me off is for example the first question, if the numbers 9 and 5 were the same I think I could do it. Do I have to get them to the same number or do I just multiply them?
  19. T

    Inverse of a logarithmic function

    Homework Statement Greetings everyone! I am having problems getting started with this question, which asks: Find the inverse of y = \frac{3-4e^x}{6-10e^x} Homework Equations The Attempt at a Solution I'm usually pretty good with these kinds of problems, but for some reason I seem to be...
  20. C

    Logarithmic Differentiation for (1+x)^(1/x): Finding dy/dx

    Homework Statement Use logarithmic differentiation to find dy/dx for y=(1+x)^(1/x). Homework Equations dy/dx The Attempt at a Solution ln y = ln (1+x)^(1/x) = (1/x) ln (1+x) (dy/dx) /y = (-1/x^2)(ln(1+x) + (1/x)(1/(1+x) dy/dx = y [(-1/x^2) ( ln(1+x) + x/(1+x) ] or (-1/x^2)(...
  21. J

    Logarithmic function as an integral

    Homework Statement I'm attempting to prove the rules for the logarithmic function using the Integral definition: Log(x)=[1,x]∫1/t dt. I think I am alright with the product rule but I'm struggling with the quotient rule: i.e. Log(a/b)=Log(a)-Log(b). I believe that I'm having trouble breaking up...
  22. J

    Logarithmic function defined as an integral

    I'm attempting to prove the rules for the logarithmic function using the Integral definition: Log(x)=[1,x]∫1/t dt. I think I am alright with the product rule but I'm struggling with the quotient rule: i.e. Log(a/b)=Log(a)-Log(b). I believe that I'm having trouble breaking up the Integral...
  23. N

    Question about Logarithmic function

    In my complex analysis textbook, there is a definition about log function, which is: log z := Log |z| + i arg z = Log |z| + i Arg z + 2kπ where k = 0,±1,±2... my question is , is there any different between log and Log, and arg and Arg? If yes, what's the different between them
  24. M

    Logarithmic differential equation

    Hi all, I have functions \eta(\mu),Z(\mu) related by \eta(\mu)=-\frac{d \ln{Z}}{d \ln{\mu}} I'm told that if we specify \eta then we have Z^{-1}(\mu)=Z^{-1}(\mu_0)\exp(\int^{\mu}_{\mu_0} dk \ \eta(k)) but upon inverting this equation, taking the log and differentiating wrt \ln(\mu) I get...
  25. C

    Help with logarithmic equations

    Homework Statement [PLAIN]http://img225.imageshack.us/img225/6162/unledsxy.png Homework Equations logb(x)-logb(y) = logb(x/y) logb(x^n) = nlogb(x) The Attempt at a Solution i am stuck on what to do because of the 4-2-2 portion. I've attempted it multiple times to get the...
  26. J

    Derivatives of the Natural Exponential and logarithmic Functions

    1. Hello there, I have a few questions on the derivative homework that I did but not sure if it's right or not. These are difficult and my teacher doesn't go over all of them so I don't know if it is right or not. Please help me so I can understand the concepts before the upcoming test. 1...
  27. E

    How Do You Calculate Elasticity from a Logarithmic Function?

    Homework Statement Find an expression (in terms of x and y) for the elasticity of y wrt x: ln(y) = 10 + 0.9x - 0.6x^2Homework Equations ln(y) = 10 + 0.9x - 0.6x^2The Attempt at a Solution I first tried the normal way of doing elasticities: ElxY = (x/y)*y'(x) which I got to: [x / 10 + 0.9x -...
  28. N

    A question in logarithmic differentiation

    In most of exercises of textbooks when ask me to differentiate f using logarithmic differentiation some time f(x) = 0 for some values of x , so I I used logarithmic Differentiation for all x in domain of f , such that f(x) not equal zero. then prove using the definition directly that for...
  29. A

    Really Simple Logarithmic Question

    Homework Statement 1. f(n) = log(n) ^ (log(n)). Simplify for Big-O notation. The Attempt at a Solution 1. I'm just trying to prove a big-O notation for one of my courses, and this was simplified to f(n) = n^log(log(n)) And I'm having a hard time seeing why. Any guidance would be...
  30. A

    Evaluating a 'logarithmic' derivative

    Homework Statement I have to evaluate the following integral: \frac{\partial \log\rho (r)}{\partial \log r} for \rho (r) = \rho_0 \Big(1+\big(\frac{r}{\alpha}\big)^2\Big)^\frac{-3 \beta}{2} where \rho_0,\alpha,\beta are constants and r is a random variable. Homework Equations...
  31. G

    Graphing of a logarithmic function

    I am facing difficulty in graphing this logarithmic function: some constant= y + log x what all steps are required to graph such a function?
  32. H

    Understanding Logarithmic Identities in Differential Equations

    This is related to differential equations, but I think my question has more to do with log identities than DE. I keep seeing equations like 1/8 lny = t + c simplified to get the solution y = ce^8t but I am unsure of the identity being used to get 1/8 into the exponent as 8. I...
  33. S

    Series expansion of logarithmic function ln(cosx)

    Homework Statement A question asks me to find the first three non-zero terms of ln(cosx) Homework Equations The Attempt at a Solution I wrote cos x as 1+(1-cos x), used the power series of ln function, expanded cosx and simplified, here is my answer: 1/2x^2 - 1/6x^4 + 1/16x^6...
  34. S

    Series expansion of logarithmic function

    Homework Statement Find first three non zero terms in series expansion where the argument of funstion is small ln(5+p) Homework Equations The Attempt at a Solution The only way I could think how to do this is by saying ln(5+p) = ln(1+(4+p)) and expanding to (4+p)-...
  35. J

    Logarithmic spiral photon orbits

    Hello, It has been said if a photon crosses the event horizon of a black hole it will have a spiral orbit; The spiral is Archimedes r = r_{i}+r_{0}\theta^2 There is one such example in "Introduction to tensor calculus, relativity and cosmology" by D F Lawden on page 165 Is it possible...
  36. romsofia

    Identities between exponential and logarithmic functions?

    Hello! I was wondering if they're any identities between exponential and logarithmic functions? Maybe identities isn't the right word, but what I'm talking about is something like euler's formula. Other than than {e^x} and {ln(x)} are inverses, if that counts. Any help is very much appreciative!
  37. D

    Calculus 1 Solving logarithmic equation

    Solve for b, f(x)=b^x Here is where I am so far lny=lnb^x lny=xlnb^x y'/y= lnb e^y'/y= e^lnb b= e^y'/y My question is, can I simplify the exponent y'/y any further, or is my answer good here? Thank you
  38. K

    Logarithmic decrement of a lightly damped oscillator

    Homework Statement The logarithmic decrement δ of a lightly damped oscillator is defined to be the natural logarithm of the ratio of successive maximum displacements (in the same direction) of a free damped oscillator. That is, δ = ln(An/An+1) where An is the maximum displacement of the n-th...
  39. dkotschessaa

    Logarithmic Differentiation Problem

    The problem I am having with these is that I'm sometimes not sure whether to begin differentiating first, or to start taking logarithms - and whether the order changes the outcome For example: Find the equation of the tangent line to the curve at the given point. y= \frac{(e^x)}{x}...
  40. Saitama

    Again a logarithmic inequality

    Homework Statement i got stuck at the question below:- Homework Equations The Attempt at a Solution I tried to solve it by simplifying it but i got stuck at:- Please help.
  41. Saitama

    Need help solving a Logarithmic Inequality?

    Got a load of Logarithmic Inequalities questions. Solved almost all of them but got stuck in a question. here's the question:- log3 |3-4x| > 2 Please help.
  42. C

    Modeling with exponential and logarithmic functions help?

    Modeling with exponential and logarithmic functions help? Homework Statement Use Newton's Lay of Cooling, T = C + (T0 - C)e-kt, to solve this exercise. At 9:00 A.M., a coroner arrived at the home of a person who had died during the night. The temperature of the room was 70 degrees F, and at...
  43. P

    Logarithmic Integral Homework: Solving S [(e^x)(e^x)]/((2+(e^x))^2) dx

    Homework Statement S = integral symbol S (e^(2x))/((2+(e^x))^2) dx Homework Equations u = 2+(e^x) (e^x) = u-2 du = (e^x) dx The Attempt at a Solution S [(e^x)(e^x)]/((2+(e^x))^2) dx S [du(u-2)]/(u^2) For the first (e^x) dx I substituted du. And since there was only (e^x)...
  44. I

    Solving Geometric Progression & Logarithmic Equations Math Questions

    I'm stuck on these three maths questions. 1) In a geometric progression, the sum to infinity is four times the first term. (i) Show that the common ratio is 3 (ii) Given that the third term is 9, find the first term. (iii) Find the sum of the first twenty terms. 2) Solve the...
  45. Pengwuino

    Medical What other senses are logarithmic?

    So many of us know that our eyes detect brightness on a logarithmic scale. Something that is twice as bright to us is in fact 2.51x as bright. So I was curious as to other senses for humans are logarithmic? The reason I was wondering is because I was wondering about how accurate you really...
  46. E

    Can a Taylor series approximation be used to solve a mixed logarithmic equation?

    "mixed" logarithmic equation Homework Statement I'm trying to solve for x in the following equation: e^(-66/x)/x^2 = c, where c is a constant Homework Equations The Attempt at a Solution By taking ln of both sides and then dividing by 2, I get to: -33/x -ln(x) = c/2 Then, in order to get an...
  47. R

    Calculating m and b for Logarithmic Regression with Small Data Set

    I have a very small set of data. Usually 3 points, sometimes 4. Best fit is a logarithmic equation y=m*Ln(x)+b How can I obtain m and b?
  48. C

    Logarithmic Differentiation for y=x^{sin(x)}

    Homework Statement Find y' using logarithmic differentiation if y=x^{sin(x)} Homework Equations The Attempt at a Solution lny=sin(x)ln(x) \frac{1}{y}(y')=sin(x)(\frac{1}{x})+ln(x)\cdot cos(x) (y')=[sin(x)(\frac{1}{x})+ln(x)\cdot cos(x)]y Did I do this one properly? Thanks in advance.
  49. H

    Another challenging logarithmic question

    Homework Statement Combine into one log log√x + 1 + ( 9log x ) Homework Equations - The Attempt at a Solution I tried converting log√x + 1 to log(x+1)^(1/2)
  50. H

    Simplify logarithmic equations

    Homework Statement Simplify log21/8 Homework Equations - The Attempt at a Solution the answer i got was -3 but it is not right.
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