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. 22990atinesh

    Approximating logarithmic series

    Can anybody tell me how this is possible
  2. D

    Can Logarithmic Properties Simplify This Equation?

    Homework Statement For clarification a have posted the equation below as a picture file.[/B] Homework Equations log(a*b)= loga + logb log(a/b) = loga-logb log(a^n) = nloga[/B] The Attempt at a Solution I don't know how to start. I can't remember the rule for powering the logarithms if there...
  3. I

    Instances where Logarithmic Differentiation doesn't work?

    Hey, In my class we just learned about logarithmic differentiation. I can see this being useful when taking the derivative of a complex function since it could be messy. But, I tried it on simpler equations as well. Everything I tried it on it seemed to work. Are there ever instances that it...
  4. N

    Calculating time in a logarithmic scale

    Hi Everyone, I have a question about logarithmic sampling. I think I might have to go and review my Z-transforms, but maybe not. The reason I am doing this is because I am automating equipment which records the data over time, but I want the sampling to be done logarithmically. If I have an...
  5. C

    MHB Logarithmic Properties of c: Exploring 0<a<1<b

    \log_{b}\left({a}\right) =c ,0<a<1<b What can you conclude about c? explain.
  6. C

    MHB Why Are My Logarithmic Equation Solutions Extraneous?

    Solve for x $\log\left({2-x}\right)+\log\left({3-x}\right)=\log\left({12}\right)$ $\log\left({(2-x)(3-x})\right)=\log\left({12}\right)$ $\left(2-x)\right)\left(3-x)\right)=12$ ${x}^{2}-5x-6=0$ $\left(x-2)\right)\left(x-3)\right)=0$ $x=2, x=3$ I have a problem with solutions because both is...
  7. L

    Kinetic Energy - use a log scale or not?

    Homework Statement Lets say we have a series of balls and an eggs. All items have equal mass but are traveling at different velocities. Thus they have differing kinetic energies. Now, for each 'ball' object we have, we want to find an egg object with a similar kinetic energy. Let's say...
  8. Q

    MHB Exponential and Logarithmic Functions

    y=CektA) First find k. [Hint:Use the given information of y=100 when t=2, and y=300 when t=4 to compute k.] B) Finally, find the value for C. [Hint use ine of the two pieces of information given in the problem to solve for C. in other words, use either y=100 when t=2 or use y=300 when 4=4 to...
  9. quantatanu0

    Cut-off Regularization of Chiral Perturbation Theory

    I was trying to learn renormalization in the context of ChPT using momentum-space cut-off regularization procedure at one-loop order using order of p^2 Lagrangian. So, 1. There are counter terms in ChPT of order of p^4 when calculating in one-loop order using Lagrangian of order p^2 . 2...
  10. Mr Davis 97

    Question on logarithmic differentiation and absolute value

    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...
  11. Mr Davis 97

    Question about logarithmic differentiation

    I have a question about logarithmic differentiation, especially concerning the absolute value involved. For example, if we have the function ##y = 2^x \sin x##, the domain is all real numbers. So what happens when we take ##\ln## of both sides of the equation? The antilogarithm must be greater...
  12. Billy.Ljm

    Integrate 1/x = ln(x) vs ln(kx)

    If k is a constant, I know \frac{d}{dx} \ln(x) = \frac{1}{x} \frac{d}{dx} \ln(kx) = \frac{k}{kx} = \frac{1}{x} However, what about \int\frac{1}{x}. I've been taught to use \int\frac{1}{x} = \ln(x), but wouldn't \int\frac{1}{x} = \ln(kx) work as well. And if this is true, there are an...
  13. L

    MHB Solving Logarithmic Equations: Expert Help with Different Bases

    Hello, I am trying to solve this, but the bases are different and I am not sure how to proceed with it... Solve the following equation. If necessary, enter your answer as an expression involving natural logarithms or as a decimal approximation that is correct to at least four decimal places...
  14. 22990atinesh

    How to Solve a Limit Problem Involving Logarithms?

    Suppose there is a limit ##\lim_{n \to \infty} \frac{n^{1.74}}{n \times (\log n)^9}## Taking logs both on numerator and denominator ##=\lim_{n \to \infty} \frac{1.74 \times \log n}{\log n + 9 \log \log n}## What can we say about the limit as n approaches ##\infty##
  15. 22990atinesh

    Small confusion regarding logarithmic formula

    I've a small confusion about formula ##\log^3 n = \log \log \log n## or ##\log^3 n = (\log n)^3 ##
  16. A

    Need some help with solving a logarithmic equation

    Homework Statement The equation is as follows: Homework Equations What technique shall I use to solve the equation? The Attempt at a Solution Tried to solve it in any way that went through my mind, including making a substitution of terms but couldn't quite get the solution or find a path...
  17. B

    Formulae with logarithmic terms

    Homework Statement w = 1/h ln(l/lo-1) w=-2.6, lo=16 and h =1.5 Find L. Homework EquationsThe Attempt at a Solution Plug in the values 2.6 = 1/1.5 ln(l/16 - 1) Make the log term the subject. ln(L/16 -1) = 1/1.5/-2.6 ln(L/16-1) = 0.666/-0.256 ln(L/16-1) = -0.256 Change the log statement to an...
  18. learn.steadfast

    Logarithmic significant figures

    In most areas of science, being able to solve and report problems to the correct number of significant figures is necessary; but I'm having trouble finding a complete set of rules for significant figures when working with non-ideal power equations. eg: I'm talking about equations requiring the...
  19. A

    MHB Evaluating a logarithmic integral using complex analysis

    Hello, I am evaluating: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ Using the following contour: $R$ is the big radius, $\epsilon$ is small radius (of small circle) Question before: Which $\log$ branch is this? I asked else they said, $$-\pi/2 \le arg(z) \le 3\pi/2$$ But in the...
  20. A

    Proving integral on small contour is equal to 0.

    Consider the integral: $$\int_{0}^{\infty} \frac{\log^2(x)}{x^2 + 1} dx$$ $R$ is the big radius, $\delta$ is the small radius. Actually, let's consider $u$ the small radius. Let $\delta = u$ Ultimately the goal is to let $u \to 0$ We can parametrize, $$z =...
  21. karush

    MHB Calculating a Logarithmic Product Series

    compute the product. $\left(\log_{2}\left({3}\right)\right)\cdot \left(\log_{3}\left({4}\right)\right)\cdot \left(\log_{4}\left({5}\right)\right)\cdots \left(\log_{126}\left({127}\right)\right)\cdot \left(\log_{127}\left({128}\right)\right)$ The answer to this is 7 I assume this can be done...
  22. Diana Dobleve

    Logarithmic derivative question

    Homework Statement 1) I am having trouble with the questions, "Use the logarithmic derivative to find y' when y=((e^-x)cos^2x)/((x^2)+x+1) Homework Equations (dy/dx)(e^x) = e^x (dy/dx)ln(e^-x) = -x ? The Attempt at a Solution First I believe I put ln on each set of terms (Though I don't know...
  23. A

    Solving for X: The Logarithmic Equations

    Homework Statement Solve for X logx-log(x+11) = -1 and log4x-log4(x+15) = -1 Homework EquationsThe Attempt at a Solution log x - log (x+11) = -1 log (x/x+11) = -1 I don't know how to solve for X after this point log4x-log4(x+15) = -1 log4 x/(x+15) = -1I don't know how to get the X out of...
  24. datafiend

    MHB Solve Logarithmic Equation: 7(3+7x)-3(4-2x)=0

    i have the equation 7(3+7x) = 3(4-2x) so, I tried 7(3+7x)-3(4-2x)=0 ...
  25. A

    MHB Challenging Logarithmic Integral: Exploring Different Methods

    Hello, The problem is $\int_{0}^{1} \frac{ln(1-x)ln(1+x)ln(1+2x)}{(1+2x)} \,dx$ Is it even worth trying? I haven't learned series yet, so if there are other methods, please let me know so I can start off. I did some differentiation under the integral sign, but it didnt work out well. If...
  26. P

    Converting Logarithmic Equations to Base 2

    How to solve this: log(base16)x + log(base4)x + log(base2)x = 7 If I have log(base16)x for example and i make 10^(log(base16)x) of it, can I transform the base 16 into an exponent? It would look like that then: 10^(logx)^16 = x^16 Would that be correct? Then, I...
  27. P

    Solving Logarithmic Problems with 0-1 Calculator

    Suppose your pocket calculator is damaged: it can only evaluate logarithms of numbers between 0 and 1. Find a way to evaluate the following logarithms with the use of such a calculator. log2 = log (10*0.2) = log10 + log0.2 = 1 + log0.2 log333 = log(1000 * 0.333) = log10^3 + log0.333 = 3...
  28. A

    MHB Logarithmic Differentiation And Exponential Growth Questions

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For the first one (part a), I went through the steps for logarithmic differentiation by using the ln laws to separate terms, and then...
  29. T

    Derivative using Logarithmic differentation

    Need to find derivative using logarithmic differentiation y = \sqrt{x(x+1)} My attempt ln y = ln \sqrt{x(x+1)} ln y = \frac{1}{2}ln x(x+1) ln y = \frac{1}{2}ln x + ln(x+1) \frac{1}{y}= (\frac{1}{2}) \frac{1}{x} + \frac{1}{x+1} \frac{1}{y}= \frac{1}{2x} + \frac{1}{x+1} \frac{dy}{dx}=...
  30. F

    MHB Solving a Logarithmic Curve: Tangent Lines & Approximations

    guys I need loads of help for this question(s)? what do i even do ?!? Sketch the curve y = lnx and find the tangent line to this curve at the point where the curve crosses the x-axis. Deduce that, for small delta, ln(1 + δ) ≈ δ . - I know what the y = Inx curve looks like but what do i do...
  31. R

    Calculating Uncertainty, involving a logarithmic quantity

    Essentially I'm asking if the uncertainty in μ = -5 log10 (d/10) is given by Δμ = 5 * ( Δd / d*ln(10) ) 1. The problem, all variables and given/known data I am to calculate the uncertainty in absolute magnitude (M), which is calculated using an equation involving logs. The equation...
  32. W

    A logarithmic convergence tests - Analysis

    Homework Statement Given a non-negative sequence \{a_{n}\}_{n=1}^{\infty}. Proove that the serie \Sigma_{n=1}^{\infty}a_{n} converge if and only if \Sigma_{n=1}^{\infty}\ln(1+a_{n}) converges. Homework Equations The Attempt at a Solution My first attempt is the direct...
  33. T

    Intersecting Functions: Solving Logarithmic Equations

    Homework Statement Determine at which points the graphs of the given pair of functions intersect: f(x) = 3x and g(x) = 2x2 Homework Equations The Attempt at a Solution I know I have to equate and solve for x so I converted them to logarithms log3x = log2x2 Don't know if...
  34. M

    Solve Logarithmic Equation: \log_{2010} 2011x = \log_{2011} 2010x

    Homework Statement \log_{2010} 2011x = \log_{2011} 2010x The Attempt at a Solution \log_{2010} 2011x = \log_{2011} 2010x \frac{\log 2011x}{\log 2010} = \frac{\log 2010x}{\log 2011} \frac{\log 2011x}{\log 2010} - \frac{\log 2010x}{\log 2011} = 0 \frac{\log 2011\log 2011x - \log 2010...
  35. S

    MHB Need help solving this logarithmic problem

    (e3x)(e4)x = e4x -15 Could use help solving this here within the next 40 minutes, appreciate the help, this is the last one out of a 75 question take home test and I'm having soo many issues. Thanks in advanced for the help.
  36. Illuvitar

    Solve Logarithmic Equation: Log _3(x-5) + Log _3(x+3)=2

    Homework Statement ##log _3\left(x-5\right) + log_3\left(x+3\right) = 2## I'm having trouble with applying properties of logarithms to solve equations, I think misunderstanding something fundamental here, I get most of the questions right but there are a handful that I have no idea what I am...
  37. W

    Electromagnetic damping logarithmic decrement task

    Homework Statement The contour consists of a condenser with capacitance C = 2.22*10^-9 F and a coil of copper wire. The diameter of the wire 5*10^-4 m, the length of the coil 20*10^-2 m. Determine the damping logarithmic decrement Λ of the fluctuations? Given answer:0.018 Homework Equations...
  38. J

    MHB Exponential and Logarithmic Problem

    I am looking for help solving for x for the question below. Any help would be greatly appreciated. 2^x=16(8^2x)
  39. A

    Why is my computer giving different answers for logarithmic laws?

    I'm using this computer program: It tells me that: log(x+1/3)=log(3x+1) Why is that so? I plug in values: they give different answers...!
  40. J

    SImplifying logarithmic functions

    [b]1. Homework Statement [/ Simplify: log base 2 x^2*y^3 Homework Equations I know that log base 2 x^2 * y^3 is log base 2 x^2 + log base 2 y^3 The Attempt at a Solution Here is what I thought: 2 log base 2 x + 3 log base 2 y But that does not seem to be simplified, it...
  41. M

    Logarithmic Mean Temperature Difference

    Homework Statement Homework Equations The Attempt at a Solution I am confused as to which streams I should use for the log mean temperature difference. I am pretty sure it should be the difference in temperature between the two streams on the same side, so I think it's (700-450)-(1100-960) /...
  42. B

    Analysing graph of a logarithmic function

    Hello everyone! I'm stuck on a small detail in a math task, and would really appreciate some help! Homework Statement Determine all local extreme points and possible max/min values for the function f(x) = x*lnx+(x*lnx)^2 where 0<x≤1/2 Homework Equations The Attempt at a...
  43. J

    Logarithmic and exponential equations

    Homework Statement (2^x - 2^-x)/3=4 Homework Equations Using log or exponential rules The Attempt at a Solution First multiply both sides by 3 so 2^x-2^-x=12 I thought I could take the log of both sides then condense the log, but that is not right. I also attempted to...
  44. P

    How to Perform Logarithmic Regression by Hand?

    I'm trying to write some code to do a regression on data weight (x) and time (y). As best as I can tell, the model should be y = b1 + b2ln(x), but I don't know how you can do this by hand (I know how to in R...). I also know how to do a simple linear regression by hand. Can it be done using...
  45. L

    Logarithmic singularities are locally square integrable

    Homework Statement I will like to show that the function f:\mathbb{R}^2\rightarrow \mathbb{R} defined by f(x)=\ln\bigg(1+\dfrac{\mu}{|x-x_0|^2}\bigg),\quad\mu>0 is in L^2(\mathbb{R}^2). Homework Equations A function is in L^2(\mathbb{R}^2) if its norm its finite, i.e...
  46. alyafey22

    MHB Logarithmic integral of second power

    This thread will be dedicated to try finding a closed form for the integral \int^1_0 \frac{\log^2(1+x)\log(x)}{1-x} All suggestions and attempts are welcomed , this is NOT a tutorial.
  47. I

    Integrating ln(√x)/x: A Challenging Logarithmic Integral

    Homework Statement ∫(ln(√x))/(x)dx Homework Equations The Attempt at a Solution I am really not sure where to start. All of the other integration problems were relatively simple, sticking with the ∫u'/udu = ln(u).
  48. C

    What is the half-life of Iodine-131?

    Homework Statement 0ops.. The title should read solving logs.. Mod note: Fixed. the original dosage contains 280 MBq of Iodine-131. If none is lost from the body, then after 6 hr there are 274 MBq of iodine-131. What is the half life of iodine I-131? Homework Equations Logcx=y The Attempt...
  49. P

    Regarding logarithmic differentiation

    Thank you for viewing my thread. I have been given the following steps for logarithmic differentiation: 1. Take natural logarithms of both sides of an equation y = f(x) and use the Laws of Logarithms to simplify. 2. Differentiate implicitly with respect to x. 3. Solve the resulting equation for...
  50. M

    How Can I Express x in this Logarithmic Equation?

    Hi, How Can I express x as a function of y in this Logarithmic equation? (Describe x by y) y=1-\frac{2x}{Ln^2(e^x+\sqrt{(e^{2x}-1)}}
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