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. C

    Lab Exercise: How do I make a logarithmic curve linear?

    Homework Statement In a certain experiment, the power (P) radiated by a light bulb filament was measured as a function of the filament's absolute temperature (T). Data: P(W) 0.45 0.95 1.8 3.5 5.6 T(K) 1000 1200 1500 1800 2000 (a) Plot the data (Done) (b) Assume a power...
  2. S

    Help Basic logarithmic differentiation question.

    Homework Statement Determine the equation of the line that is tangent to y=8^x at the point on the curve x=1/2. Homework Equations Differentiation Rules m=y2-y1/x2-x1 The Attempt at a Solution y=8^x dy/dx=8^x(ln8) =8^0.5(ln8) =5.9 y=8^x =8^0.5 =2.8 5.9=y-2.8/x-0.5...
  3. V

    Solving Logarithmic Equation: Log81x + logx3 = 2

    The equation is the following: Log81x + logx3 = 2 I've been bashing my head with change of base but i can't seem to get anywhere with it. 81 is rewritable as 34 which seems like it has something to do with the solution, but i just can't seem to get the hang of it. I always get stuck with...
  4. T

    Solving Logarithmic Equations Involving x & y

    Hi :) This is my first time using this forum, so thanks for your help in advance. Homework Statement Use the laws of logs to write y as a function of x for each. Then state domain. a) log(xy) = 2 log(x-3) b) log(y) + 3 = log(y + 1) + log(x) c) log (x2/y) = 2 log(x + 5) Answers: a)...
  5. P

    Tricky logarithmic equation at International Baccalaureate high school

    Hi all! From time to time, I work as a math tutor. Normally this doesn't cause me any trouble, but a couple of days ago a student from an IB school came to me with this problem: 2*5(x+1) = 1 + 3/(5x). Solve for x and write the solution on the form a + log5b. I've been trying to solve...
  6. O

    How does the chain rule apply to logarithmic differentiation?

    Hi everyone - I'm in my first year at uni, and I was given a derivative that i don't quite get: y(x) is a function of x then y` = (dy/dx) d/dx (ln y(x))=1/y(x)*(dy/dx) =(y`/y) This is given by the chain rule. I don't understand this step... Any help? Thanks in advance, Owen.
  7. J

    Logarithmic Derivative of y = 5ln(7lnx)

    Homework Statement y = 5ln(7lnx) Homework Equations y' = 5 x 1/7lnx +97lnx0 x 1/x = 5(7lnx)/7lnx + 5ln x 7/x The Attempt at a Solution y' = 5 + 35ln/x is this right?
  8. H

    Derivatives using Logarithmic Differentiation

    Homework Statement Using logarithmic differentiation calculate the derivative of y=e^(x^x) The Attempt at a Solution y=e^x^x LNy=LNe^x^x LNy=x^xLNe ... Stuck! This seems to be the only way you can do it, but once I get to that part I'm not sure what else there is to do. I...
  9. I

    Mathematica Logarithmic 3D plot in Mathematica

    Hi all, Does anyone know how to get Mathematica make a 3D plot with logarithmic axes? I need a, say, 3D version of LogLogPlot. I have a 3D function Rdb[nf, pf] which I want to plot for {nf, 10^10,10^18} and {pf, 10^10,10^18}. I can get cross sections of what I need when I do...
  10. M

    Simple algebraic / logarithmic question

    Homework Statement (1.024)^2 = (1 + m)^12 Homework Equations The Attempt at a Solution 1.048576 = (1+m)^12 I tried to do a square root of 12 for both sides but does not appear to be correct. I've also tried doing log to bring the exponents to the front for both sides, but...
  11. E

    Logarithmic Inequalities: Solving e^(2-3x)>4

    Homework Statement Solve the inequality: e^(2-3x)>4 Homework Equations none The Attempt at a Solution would i start of like this? e^(2-3x)>4 ln(2-3x)>ln4 if so how do i continue?
  12. P

    Proof of logarithmic properties.

    Homework Statement There are two log properties that I have to prove: 1) Explain why ln(b1/n)=(1/n)ln(b) for b>0, set b=an 2) Explain why ln(ar)=rln(a) for any r in Q and a>0, ie r is rational. Homework Equations ln(an)=nln(a) The Attempt at a Solution In a previous question...
  13. M

    How do logarithms convert numbers like 8.7 and -2.3 to whole numbers?

    I am unsure of how they were able to change 8.7 to 8 and make -2.3 to 2. Whats the explanation behind this?
  14. E

    Q1 - Equation with fractional and logarithmic exponents

    Homework Statement 4 1/(x+1) · 8 1/(x+2) = 9log3 2 Homework Equations n/a - same as 1 The Attempt at a Solution To get rid of the exponents and be able to solve for x, I first took the logarithm with base 4 of all terms: log44 1/(x+1) · log48 1/(x+2) = log49 log3 2 This equals: 1/(x+1)...
  15. M

    Calculus Logarithmic Functions help please

    Calculus Logarithmic Functions help please! The question is: A particle moves alonge the x-axis with position at time t given by x(t) = e^(-t) sin t for 0 ≤ t ≤ 2π. 1. Find the time t at which the particle is farthest to the left. Justify your answer. 2. Find the value of the constant A...
  16. 1

    Solving Logarithmic Equations: 2 Problems

    Homework Statement I have 2 problems ,which are: 1) Solve :log 7 (x 3+27) + log 7 (x+3) 2) Solve for x:log5 ( 5 1/x+ 125) log 5 6+1+1/2x Homework Equations The Attempt at a Solution The first one tried like this : log 7 { (x 3+27)/(x+3) }= 2 or 7 2 = (x 3+27)/(x+3) But...
  17. H

    Using search engine logarithmic scales

    Hi there, I have a problem for work that is stirring up lots of memories of University Math courses, and I need help! I work in search engine optimization, and search engines use logarithmic scales to value pages and websites. I'm using a tool that mimics those scales and assigns a number...
  18. M

    Solve Logarithmic Integral int. (ln(ax+b))^2

    Does anyone know how to solve this integral? int. (ln(ax+b))^2 Struggling! Thanks :D
  19. DaveC426913

    What is the relationship between depth and pressure in water?

    I'm discussing decompression in shallow water. I never studied logarithms. At sea level, a diver is at 1 atm, at 33ft: 2atm, at 66ft: 3atm. How does it work between 33 and 0? Specifically, at what depth is 1.25atm? Is it less than 8.25ft or more than 8.25ft?
  20. W

    Evaluating/Solving Logarithmic expressions/equations

    Homework Statement Evaluate the expression "log base 8 of 16" Homework Equations Don't know of any The Attempt at a Solution I don't know where to begin. I have read the textbook chapter on logarithms three times over, and I am now more thoroughly confused than ever. My textbooks...
  21. B

    The Fire Tower - Logarithmic spiral

    Homework Statement I'm trying to figure out what exactly this equation (the one inside the red box) means for this article. The whole article is the attached PDF. Homework Equations The Attempt at a Solution I've been looking at this article for 2 days now and I'm doing OK for the...
  22. J

    Simple problem: logarithmic decrement

    Homework Statement Note this is exam revision rather than actual course work worth marks, so there is no need to be deliberately vague :) The question comes in two parts, regarding a lightly damped harmonic oscillator with frequency 10 kHz and an amplitude that decays by 25% over 300...
  23. majormuss

    Solving Logarithmic Equations: Step-by-Step Guide

    Homework Statement My book has these steps for evaluating a logarithmic equation and I don't quite understand what is going on. It says... log (2) x(x-4)=5 x(x-4)=2^5 <---------> this is the level I am getting problems... how does log(2) divide the 5 on the other side and end up getting...
  24. I

    Line integral & logarithmic spiral

    Homework Statement Calcualte the value of \int\limits_L \sqrt{x^2+y^2}dl, where L is an arc of a logarithmic spiral r=ae^{m\phi} between points A(0,a) and B(-\infty,0). Problem: I can't find a value of \phi where x=-\infty or y=a. Homework Equations We parametrise and get...
  25. E

    Solving Logarithmic Equations with Given Logs

    Homework Statement 1. Use the fact that loga(10)= 2.4197 and loga(13)=2.6955 to solve the following a. loga(130) b. loga(1.3) c. loga(1000) Homework Equations The Attempt at a Solution I'm not really sure how to do this i missed yesterdays class i don't really want...
  26. J

    Solving for Logarithmic Decrement in LCR Circuits

    Homework Statement Basically, I have LQ''(t) + RQ'(t) + (1/C)Q(t)=0, and I'm supposed to "Show that the ration of the charge Q between two successive maxima is given by exp(RTd/2L), where Td is the time between two successive maxima. The natural logarithm of this ration is called the...
  27. C

    Differentiating logarithmic functions

    Homework Statement Differentiate the following: y = e4x/x2+1 2. The attempt at a solution I know you have to use quotient rule here. so I wrote out (x2+1)* d/dx e4x) - d/dx (x2+1)* e4x all over (x2 + 1) 2 I have no idea how to continue on from here...how would you...
  28. T

    Simplifying Logarithmic Expressions to \log_x{y}

    Homework Statement Simplify (\log_a{b})(\log_b{c})(\log_c{d}) in terms of \log_x{y} Homework Equations The Attempt at a Solution well , the furthest i can reach is \frac{\log_xd}{\log_xa} , how do i express in terms of \log_x{y}
  29. P

    Evaluating Logarithmic Integral with Cosine and Sine Integrals

    I was trying to evaluate this integral, \int\frac{dx}{\ln x} I substituted x=e^{i\theta} and I get, \int\frac{e^{i\theta}}{\theta}d\theta which is, \int\frac{\cos \theta}{\theta}+i\frac{\sin \theta}{\theta} \ d\theta \int\frac{\cos \theta}{\theta} \ d\theta+i\int\frac{\sin...
  30. Z

    Renormalization of Logarithmic divergences

    how can logarithmic divergences be renormalized ? for example if i have \int_{0}^{\infty} \frac{log^{n}(x)dx}{x+a} differentiation with respect to 'a' and integration over 'x' gives finite result for example \int_{0}^{\infty} \frac{dx}{x+a}=-log(a)+C here 'C' would be an extra...
  31. C

    Reversing Logarithmic Integral Function

    The logarithmic integral function, which is what you get by integrating 1/ln(x), is closely linked to prime numbers. It approximates the number of primes smaller than x. Heres an infinite series which describes the function: li(x)=\gamma+ln(ln(x))+\sum^{\infty}_{n=1}\frac{ln(x)^n}{n*n!}...
  32. E

    Interperating a graph with logarithmic scales

    Homework Statement see attachment Homework Equations log10x=y 10y=x The Attempt at a Solution Since I'm rather confused on reading a logarithmic scale, I thought I'd post this to see if I'm doing this right and/or if there's a better way to do the problem. a) T=500 C using the...
  33. D

    Opamp logarithmic & exponential amplifiers

    So I was playing around with logarithmic & exponential amplifiers in my lab class. I was looking at the following equations: http://upload.wikimedia.org/math/7/7/6/77663157d5b97ceb2e3edac5f587a620.png and http://upload.wikimedia.org/math/b/3/c/b3c569c85552561e41dec916f6e8ebe8.png...
  34. J

    Linear to logarithmic scale conversion

    Hi! I'm making a computer program that represents some quantities in a graph in this way: x'i=(xi-xmin)/(xmax-xmin) so that the possible values of x range from 0 to 1. This is a linear scale. I want to do the same with the logarithmic values of xi. That is, I want to implement a log...
  35. A

    What is the error in this simple logarithmic equation?

    Can anyone please tell me what is wrong with this solution log(5x-17) = log(4x-14) As logarithmic functions are one-one so, 5x-17 = 4x-14 => x =3
  36. srfriggen

    Logarithmic vs exponential scales

    can someone please explain the difference. Graphically and mathematically it is easy to see they are inverses. But I see certain scales like the Richter scale that seem to increase exponentially, but are labeled as logarithmic scales. for example, on the richter scale with each increase in...
  37. M

    Derivative of a logarithmic function

    Homework Statement Compute the derivative: d/dx log base 10 (x^3 + x^2) Homework Equations The Attempt at a Solution (3x^2 + 2x)/(ln 10)(x^3 + x^2) seems a little off.. not sure though thanks for the help in advance! (:
  38. C

    How do I solve for x in this logarithmic equation?

    Homework Statement logx13-3log13x=-2Homework Equations n/a The Attempt at a Solution (log13/logx)-(3logx/log13)=-2 What do I do next?
  39. C

    Please solve for x in this logarithmic equation? 4^x + 2^(x+1) = 60

    Homework Statement 4x + 2(x+1) = 60 Homework Equations N/A The Attempt at a Solution 4x + 2(x+1) = 60 22x + 2(x+1) = 60 I don't know where to go from there.
  40. R

    Limit of logarithmic functions

    lim_{i\rightharpoonup\infty} \frac{ln(4^{i}-1)}{ln(2^{i})} If I set this up right it should go to 2, but I'm pretty rusty and every time I try to work this out I end up getting garbage or repeating behaviors that I can't do anything with... Anyone know what exactly to do with it? edit: Ack...
  41. L

    Domain of logarithmic funtion

    6throute (x-7) -------------- log11(13-x) What is the domain of the function in interval notation. I know to set x-7 > 0, so that gets me x>7 , and I know I have to set 13-x > 0, which gets me x < 13. But the notation that is correct is [7,12)U(12,13). Where do you...
  42. M

    Simplifying the Derivative of a Complex Rational Function

    find derivative of y=(sqrt(8x^4-5))/ (x-1) ok...after working out the tricky calculations i get for my final answer: (32x^4sqrt(8x^4-5)-64x^3sqrt(8x^4-5)) / (16x^6-32x^5+ 16x^4-10x^2+20x-10) I don't know if you want to do the math...im just wondering if i can simplify it anymore. thanks
  43. K

    Use the properties of logarithma to xpand the logarithmic function

    Homework Statement Use the properties of logarithma to xpand the logarithmic function ln[(x2+1)(x-1)] Homework Equations The Attempt at a Solution [ln x2+ln 1]+[ln x-ln 1] 2 ln x+ln x
  44. S

    Logarithmic Help: Solving for z, m & n

    Homework Statement http://img20.imageshack.us/img20/1476/mathproblemi.jpg Homework Equations Properties of logs. The Attempt at a Solution For the first two I did: Int( \frac{Log(2)}{Log(1.1)} ) Doing this resulted in the correct answers for the first two boxes. When I...
  45. jacksonpeeble

    Derivative of Exponential and Logarithmic Function Concept Question

    Homework Statement y=9-x Homework Equations The Attempt at a Solution dy/dx=-9-xln9 We are provided with the answers. I do not understand the concept behind how to get from the question to the solution. What rule(s) are applied? If somebody could solve this single question...
  46. M

    Complex analysis - the logarithmic function

    Homework Statement Show that the function Log(-z) + i(pi) is a branch of logz analytic in the domain D* consisting of all points in the plane except those on the nonnegative real axis. Homework Equations The Attempt at a Solution I know that log z: = Log |z| + iArgz + i2k(pi)...
  47. F

    Simplifying logarithmic expressions

    simplify the following expression log2 4^3x-1 (2 is a subscript of log and 3x - 1 is a superscript of 4) i can rearrange this to 4log2 3x - 1, i am thinking after this it becomes 4 log3 + 4 logx, but i don't know what to do with the -1 please show all the steps and explain.
  48. S

    Derivatives: Logarithmic Function help

    Homework Statement Find an equation of the tangent line to the curve $\displaystyle \Large y = (2 x^2+5 )\ln (4 x^2-3 )+7$ when x = 1. Homework Equations $\displaystyle \Large y = f'(x_0)(x-x_0)+f(x_0).$ The Attempt at a Solution the fact that the tangent line to the curve y =...
  49. L

    Tricky Logarithmic Indefinite Integral

    Homework Statement Hi, Our professor has only taught us these methods for Integration...thus far: Direct Integration Substituion Method So theoretically we should be able to solve this problem without using Integration by parts or partial fractions...: Integrate...
  50. T

    How do I solve for the integral of 1/(xlnx) using logarithmic integration?

    Homework Statement \int^{e^2}_{e} \frac{1}{xlnx} dx Homework Equations The Attempt at a Solution I substituted U= xlnx So DU= (\frac{1}{x}dx...so Du * X = 1dx From there I am stuck!
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