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. brotherbobby

    Solving a nested logarithmic equation

    Problem statement : Let me copy and paste the problem on the right as it appears in the text. Solution : Using the Relevant Equations (2) and (3) above, we can claim that \begin{align*} &\log_{2x^2+3x+5}(x^2+3)=1\\ &\Rightarrow x^2+3 = 2x^2+3x+5\\ &\Rightarrow x^2+3x+2=0\\ &\Rightarrow...
  2. chwala

    Solve for ##x## in the given logarithmic equation

    ##\frac{1}{log_x 2}##+ ##\frac{1}{log_x 3}##+##\frac{1}{log_x 6}##=## 3.6## ##log_2x + log_3x+log_6x =3.6## ##log_2x ##+##\frac{log_2x}{log_2 3}##+##\frac{log_2x}{log_2 6}##=##3.6## ##log_2x ##[1+ ##\frac{1}{1.58496}##+##\frac{1}{2.58496}]##=##3.6## hmmmm it took me some time here to note that...
  3. D

    B Is Mean Temp in 2 phase Heat Exchangers Higher Than Logarithmic Mean?

    Is Mean Temp in 2 phase Heat Exchangers Higher Than Logarithmic Mean? I am looking at this paper: https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1314&context=iracc Having a bit of trouble understanding all of it, but my basic question is just: If I have a heat pump where the hot...
  4. FEAnalyst

    I Logarithmic scale - interpolation

    Hi, knowing the coordinates of two points: ##(x_1,y_1)## and ##(x_2,y_2)## on a linear scale plot, I can use linear interpolation to get ##y## for a point of known ##x## using the formula below: $$y=y_1+(x−x_1) \frac{(y_2−y_1)}{(x_2−x_1)}$$ But how does it look like in the case of logarithmic...
  5. J

    MHB (ask) Calculating Logarithmic Question

    May I know how to calculate this question without a calculator? The final answer of this question is 3 but I really have no idea how to work on it to get the final answer.
  6. OwlsInATrenchcoat

    B Logarithmic IV graphs of diodes

    Hello there, I've been working through a task (that doesn't have an answer sheet or explanation) in which we plot I against V for three different diodes. Each has a different threshold voltage and displays the usual charcteristic curve. The final question is this: "It is suggested that the...
  7. Monoxdifly

    MHB [ASK] Logarithmic Equation

    A friend asked me how to solve this question: log_2(x+2)+log_{(x-2)}4=3 I said I had no idea because one is x + 2 and the other one is x - 2. If both are x + 2 or x - 2, I can do it. He said that if that's the case, even at his level he could solve it. This is what I've done so far regarding the...
  8. T

    B Why is an inverse logarithmic scale chosen for the magnitudes of stars?

    Star magnitudes of brightness seem to use inverse logarithmic scales, is there a benefit to this? Why was this chosen, i can understand logarithmic might make it easier to interpret data in same way we do similar for earthquakes etc. But why inverse ? When i look at a HR diagram for example (...
  9. S

    B Confusion about the domain of this logarithmic function

    Should I just follow the original question? If given as ##f(x)=\ln x^4## then the domain is x ∈ ℝ , x ≠ 0 and if given as ##f(x) = 4 \ln x## the domain is x > 0? So for the determination of domain I can not change the original question from ##\ln x^4## to ##4 \ln x## or vice versa? Thanks
  10. D

    I Logarithmic terms in a system of equations

    (I hope this is not a double posting) I want to solve this system of equations, containing logarithmic terms: ##7\ln(a/b)+A = 7\ln(d/e)+D = 7\ln(g/h)+G## ##7\ln(a/c)+B = 7\ln(d/f)+E = 7\ln(g/i)+H## ##7\ln(b/c)+C = 7\ln(e/f)+F = 7\ln(h/i)+I## ##a\phi_1+d\phi_2+g\phi_3=X##...
  11. scottdave

    B Using a logarithmic scale to represent COVID-19 growth

    The author, John Burn-Murdoch, shows here ( https://threadreaderapp.com/thread/1237748598051409921.html ) how the logarithmic scale can give a better "sense" of what is happening. In linear scales, some countries' data is squashed to almost nonexistent, while others explode out of control. I...
  12. JD_PM

    Central force on a particle following a logarithmic spiral

    I want to focus this question on understanding the force ##F(r)## I get (thus, I want to focus on c) ). However, below the dashed line, I included steps on how I derived ##F(r)##. We are going to work in polar coordinates. Knowing that the acceleration is: $$a = \Big( \ddot r - r \dot...
  13. S

    Solving for y in a logarithmic equation involving |y|

    Integrating both sides of the equation yields ##\ln{|y|}=-\frac{1}{4x^2}+C ## ##\iff \ln{|y|}=-\frac{1}{4x^2}+\ln{D} ## ##\iff |y|=De^{-\frac{1}{4x^2}}## At ##(1,1)##, ##D=e^{\frac{1}{4}}##. So for ##y>0##, ##y=e^{ \frac{1}{4}-\frac{1}{4x^2}}##, and for ##y<0##...
  14. karush

    MHB 242 Derivatives of Logarithmic Functions of y=xlnx-x

    $\tiny{from\, steward\, v8\, 6.4.2}$ find y' $\quad y= x\ln{x}-x$ so $\quad y'=(x\ln{x})'-(x)'$ product rule $\quad (x\ln{x})'=x\cdot\dfrac{1}{x}+\ln{x}\cdot(1)=1+\ln{x}$ and $\quad (-x)'=-1$ finally $\quad \ln{x}+1-1=\ln{x}$...
  15. V

    MHB Rewrite in logarithmic form: e^(-1) = c

    Rewrite in logarithmic form: e^(-1) = c
  16. L

    MATLAB How to transform a plot to use a logarithmic scale?

    I wrote the following code in MATLAB: t = [0:0.001:0.1]; noise = randn(1,size(t,2)); a = 15*10^9; b = 15*10^(-3); c = 7*10^8; y = a*exp(-t/b)+c+noise*100000000; fun = @(p,t)p(1)*exp(-t/p(2))+p(3); p0 = [15.5*10^9, 14*10^(-3), 6*10^8]; p = lsqcurvefit(fun, p0, t, y); t_fit = [0:0.0001:0.1]; y_fit...
  17. YoungPhysicist

    Determine whether a logarithmic function is odd or even

    Homework Statement Determine function $$f(x) = \log(\sqrt{x^2+1}+x)$$ Is odd or even. Homework Equations ##\log(a+b) = \log(a) + \log(1+b/a)## The Attempt at a Solution First I thought it is a even function without considering the x at the end, which of course isn't the actual case. Then I...
  18. Y

    MHB Logarithmic Equation solve log_(3x)3+log_(x/3)3=5/12

    Dear all, I wish to solve the following logarithmic equation: \[log_{3x}3+log_{\frac{x}{3}}3=\frac{5}{12}\] My intuition was to start with changing the base of both logarithms to 10 (or any other number), but couldn't continue from there. Can you assist please ? Is there a meaning to the fact...
  19. PlasMav

    E' vs. E_2 Neutron Scattering and Logarithmic Energy Loss

    Hello, I just had a little debate with my professor after taking my final exam. He had given us an additional formula sheet at the last second (hand written on the projector) which confused me. The question was a 7 MeV neutron collides with several U-238 atoms before reaching 2 MeV. How many...
  20. opus

    Logarithmic Differentiation Problem

    Homework Statement Find the derivative of ##f(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}## 2. Homework Equations The Attempt at a Solution Please see attached image. The given solution has a 4 in the numerator, and I don't see how I can come across that with the way that I'm going. I think I'm making...
  21. GeertTimmerman

    Magnetic field and logarithmics

    Hello everyone! I'm in my first year of my bachelor in physics and astrophysics at Leiden University, The Netherlands. We have to perform a small experiment were we use a transducer to measure a logarithmic correlation between two certain quantities. As an example our prof. said you could...
  22. R

    MHB Finding the Domain of a Logarithmic Function f(x) = log_5(8 - 2x)

    I need to find the domain of f(x) = log5(8 - 2x). I am not sure if I could find this by doing a inequality equation, but I think my professor wants us to do it on a number-line anyway. I'm sorry, but I am not sure where to start.
  23. opus

    Solution check for Logarithmic equation

    Homework Statement Solve: ##ln(x-4)-ln(x)=6## I know the solution is no solution, but the value I got is what I am unsure of as I could have got a million different values for x that would result in no solution, but I'm not sure if the value that I got is correct. Once you hit the "no...
  24. opus

    B Question about a Logarithmic Property

    Say I have ##log_5(x)=log_5\left(\frac{2x+3}{2x-3}\right)## This means that the value of the LHS and RHS are equal. I take this to mean that "5 raised to some exponent is equal to both x and ##\frac{2x+3}{2x-3}##. I can now write this as ##x=\frac{2x+3}{2x-3}## because since the function is...
  25. opus

    B Characteristics of the parent logarithmic function

    I just started going over logarithmic functions in my text, and I have a question on a summary it gives on the parent function ##f\left(x\right)=log_{b}\left(x\right)## In the attached image, it says that "for any real number x...we see the following characteristics of...
  26. dave202

    B Has anyone seen this logarithmic spiral creation before?

    <Moderator's note: Image added because otherwise the thread might once become unreadable.> I have reason to believe this could have applications in physics, but right now it's just a mathematical result I came across recently. Either way, I think it is very interesting and fun to look at. This...
  27. alijan kk

    Logarithmic integration

    1. The problem statement, a ll variables and given/known data Homework Equations The Attempt at a Solution what is t equal to here , how should i think it ?
  28. D

    MHB Finding equation from logarithmic graph

    I've spent some time researching and trying to find an equation for this line, but it's not exact. I'm only searching for the equation of the line that descends towards zero (the angled line). I plugged in some numbers and it does not match the graph, the line on the graph is steeper. I start...
  29. T

    MHB Logarithmic Functions: Solving Questions & Finding Carrying Capacity

    Hey guys, I have a couple of questions here. One, I was just wondering if someone could elaborate on, and the second, I worked it out, but more by guessing. I was hoping someone would be able to help explain both. Here is the first of the two questions So, part a was fairly straightforward...
  30. W

    B Is log8 (x/2) same as log8 x/2?

    Is log8 (x/2) same with log8 x/2?
  31. P

    Prove that the logarithmic function is continuous on R.

    Homework Statement Prove that f\left(x\right)=\log_{a}x is continuous for all \mathbb{R}. Homework Equations [/B] I must find a \delta>0\in\mathbb{R} for a given \varepsilon>0 such that \left|x-x_{0}\right|<\delta\Rightarrow\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon. The Attempt at a...
  32. E

    B Logarithmic growth vs exponential growth

    From the book Calculus made easy: "This process of growing proportionately, at every instant, to the magnitude at that instant, some people call a logarithmic rate of growing." From Wikipedia: "Exponential growth is feasible when the growth rate of the value of a mathematical function is...
  33. onemic

    Exponential and logarithmic Equation Problems

    Homework Statement Evaluate each of the following expressions without using a calculator. 1) log216√8Solve for the unknown value in each of the following equations without using a calculator. 2) 3(x+4)−5(3x)=684 3) 7(42x)=28(4x) Homework Equations Exponent law for multiplication The...
  34. S

    A Logarithmic divergence of an integral

    I would like to prove that the following integral is logarithmically divergent. $$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$ This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##...
  35. terryds

    B Strange logarithmic property

    I encountered this in http://calcchat.com/book/Calculus-10e/8/4/7/ How come the above expression equals the below? What I know it should be 4 ln(x/(4+sqrt(16-x^2))) which means the -1 becomes the power of that thing inside ln. Please help me. I really don't get it.
  36. Jules Winnfield

    I How do I compare a model to logarithmic data?

    I have a model which is quadratic (e.g. ##y = k x^2##). I'm comparing it against a large set of data (galaxy cluster masses) which spans several Log10 decades (e.g. ##10^{11}## to ##10^{15}## solar masses). What is the right way to say how good the data fits the model? Obviously the errors in...
  37. dfklajsdfald

    Finding the Value of axb on the Unit Circle | Round to the Nearest Thousandths

    Homework Statement the point (log a, log b) exists on the unit circle. find the value of axb. round to the nearest thousandths. Homework Equations x2 + y2 = 1 The Attempt at a Solution x2+y2 = 1 loga2+logb2 =1 2loga+2logb = 1 2(loga+logb) = 1 loga + log b = 0.5 logb = 0.5−loga now i try...
  38. R

    B Logarithmic Function: Can Domain of Logarithm be R?

    can domain of logarithm function be R . i think it can and the same time it can't it can like log(x2) but at the same time i think all the logarithm function should be one to one function
  39. D

    How to find this answer using logarithmic table?

    [Mod Note: moved into homework forum, so template not present] 3/5840=0.00051 How to find this answer using logarithmic table? Thanks in advance.?
  40. A

    Logarithmic scale for the Laser Intensity.

    I drawed a relation between the growth rate of the material to the laser intensity. After drawing, My professor told me that I must convert both values of the growth rate and the laser intensity to their values in the logarithmic scale. I don't know how. May I get a help.
  41. Mr Davis 97

    B Solving this logarithmic equation analytically

    I have the equation ##3x + \log_5x = 378##. Is there an analytical way to solve for x? Or for this equation are we forced to just try possible values, such as powers of 5?
  42. ubergewehr273

    A problem about logarithmic inequality

    Homework Statement If a,b,c are positive real numbers such that ##{loga}/(b-c) = {logb}/(c-a)={logc}/(a-b)## then prove that (a) ##a^{b+c} + b^{c+a} + c^{a+b} >= 3## (b) ##a^a + b^b + c^c >=3## Homework Equations A.M ##>=## G.M The Attempt at a Solution Using the above inequation, I am able...
  43. amrmohammed

    Logarithmic mean temperature difference for heat transfer

    Homework Statement [Update: just realized that the LMTD is a temperature difference, so my question was not valid] :P Calculate the logarithmic mean temperature difference (LMTD) to heat water flowing through a tube from 21 C (Ti) to 40 C (Te) if the tube has a fixed temperature of 45 C (Ts)...
  44. A

    I Plotting number of halos per logarithmic bin

    Hello all,. I'm right now playing the publicly released data from the Illustris simulation. Im trying to plot the number of halos per logarithmic mass bin or the number of halos between mass M and M+dM I'm only familiar using theoretical models like Press-Schecter and I have no idea how do...
  45. ruivocanadense

    I Ln x as a logarithmic function

    My book finds a function of x say ln(x). It is the area under 1/x. Having the properties (d/dx) ln x = 1/x and ln 1 = 0. It says it determines ln(x) completely. It satisfies the laws of logarithms, but why can I regard it as a logarithm just because it satisfies those laws?
  46. Brandon Trabucco

    B Complex Integration By Partial Fractions

    Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...
  47. P

    Ranges of Projectiles With and Without Drag

    I recently completed a lab using an online projectile simulator about the range of projectiles. I launched a projectile with different initial speeds (5 m/s, 10 m/s, 15 m/s, 20 m/s, and 25 m/s). For each trial, I did the launch with and without air resistance and I plotted Range (with Air...
  48. T

    SEM involving a logarithmic

    Hi, this isn't exactly a homework question, but this seemed like the most appropriate place to put it. Homework Statement I have an equation in the form: log(a)=log(b)+c. I also have standard errors (SEMs) for b and c. I want to find the standard error for log(a) (i.e. log(a) +/- E(log(a)))...
  49. J

    MHB Logarithmic Integral Calc: $$\int_{0}^{1}\ln\left(1+\sqrt{1-x^2}\right)dx$$

    Calculation of $$\int_{0}^{1}\ln\left(1+\sqrt{1-x^2}\right)dx$$ I Have tried like this way:: Let $$I = \int_{0}^{1}\ln\left(1+\sqrt{1-x^2}\right)dx$$ Put $x=\sin \theta\;,$ Then $dx = \cos \theta d\theta$ and changing limits, we get $$I = \int_{0}^{\frac{\pi}{2}}\ln\left(1+\sin \theta...
  50. S

    Oribit integrator for a logarithmic potential

    Hello! Right know I'm trying to make an orbit integrator for solving a logarithmic potential with the form: \begin{equation} \Phi= \frac{v_0^2}{2} ln(x^2+ \frac{y^2}{u^2} + r_0^2) \end{equation} where v0, u, and r0 are constants My approach is to use, \begin{equation} \ddot{q} =...
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