Log Definition and 632 Threads

In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Monomials – relationships of the form



y
=
a

x

k




{\displaystyle y=ax^{k}}
– appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used.

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  1. Z

    Approximating Arctangent and Natural Log

    I have posted this on other forums, and I have discussed this with my professors, but I thought I would share it here for those interested. Essentially, I have a function that efficiently approximates arctangent on [-1,1] and ln(1+x) on [0,1]. For some background about me, I am a Z80...
  2. Z

    What is the domain of y = log [(4-t)^(2/3)] ?

    Hello to all: I was trying to solve the following question: what is the domain of the function y = log [(4-t)^(2/3)] for Real t. Here is my reasoning: The domain of the log(x) is x>0. I would have interpreted this question in terms of the log of the cubic root of (4-t)^2. Thus, I was...
  3. J

    Is there a relation between log and arcs for complex numbers?

    If there is a formula relating the exponential with sine and cosine normal and hyperbolic (exp(ix) = cos(x) + i sin(x), exp(x) = cosh(x) + sinh(x)), there is also a formula relating the logarithm with arcsin, arccos, and arcsinh arccosh?
  4. Y

    Distribution of Log of Variance

    Homework Statement If Y_1, Y_2, ... are iid with cdf F_Y find a large sample approximation for the distribution of \log(S^2_N), where S^2_N is the sample variance. Homework Equations The Attempt at a Solution The law of large numbers states that for large N S^2_N converges in...
  5. D

    How do I plot a log log graph using this equations?

    (HgAh)=ChG^mL^n (HdAm)=CmG^pL^q I need to plot (HgAh) and (HdAm) As Fuction of G( Gas flow rate) with lines of Constant L( Liquid flow rate) and discern M and P from the equations above from Log Log Graphs. I have collected a table of raw data with Various G at constant L. How do I...
  6. G

    Cat walking on a log floating on water

    Homework Statement A dog and a cat, are standing at opposite ends of a uniform log that is floating in a lake. The log is 11.0m long and has mass 460kg . The dog has mass 9.0kg and the cat has mass 7.0kg . Initially the log and the two pets are at rest relative to the shore.The cat then begins...
  7. N

    How Do You Start Solving a Logarithmic Equation?

    Homework Statement Solve the logarithmic equation for x: Log2 (x^2 - x - 2) = 2 Homework Equations The Attempt at a Solution I'm just totally lost on how to even start...
  8. E

    What Is the Expectation E(log(x-a)) for a Log-Normally Distributed Variable x?

    What is the expectation, E(log(x-a)), when x is log normally distributed? Also x-a>0. I am looking for analytical solution or good numerical approximation. Thanks
  9. A

    Solve Log Law Problems Homework Statement

    Homework Statement Write as a single logarithm:Homework Equations Logarithm Laws: log_a(xy) = log_a(x) + log_a(y) log_a(\frac{x}{y}) = log_a(x) - log_a(y) ___________ Problem Set: log_{10}A + log_{10}B - log_{10}C \frac{1}{2}logX - 2log4 2logN + 3logX The Attempt at a Solution I...
  10. F

    MHB Do you need absolute value around argument for log and ln?

    When I learned about derivatives I was taught to put the absolute value sign around the argument for ln and log. For example \log{|x|} and \ln{|x|} instead of log(x)ln(x). Does this make a difference? Should both brackets and the straight lines be used? When taking the derivative what is the...
  11. M

    Natural log limits as n approaches infinity

    My question is: Show the limit of x_{n}=\frac{ln(1+\sqrt{n}+\sqrt[3]{n})}{ln(1+\sqrt[3]{n}+\sqrt[4]{n})} as n approaches infinity Solution: {x_n} = \frac{{\ln (1 + {n^{\frac{1}{2}}} + {n^{\frac{1}{3}}})}}{{\ln (1 + {n^{\frac{1}{3}}} + {n^{\frac{1}{4}}})}} = \frac{{\ln \left(...
  12. Jameson

    MHB Log likelihood function and EM algorithm

    I'm going to start by asking about an example from class and then hopefully use that to work on the problem I need to solve. Here is an example: Let's say we have a multinomial distribution $x \sim M(n;.5+.25\theta,0.25(1-\theta),0.25(1-\theta),0.5\theta)$. The likelihood function of $\theta$...
  13. M

    Equivalence of Log and Argument for Complex Functions?

    Hello! So I have questions on this equivalence: Imlog[(1+x)/(1-x)] = arg [(1+x)/(1-x)] where x: complex number How is this true? Is it always applicable no matter what form of complex function is under calculation? Thank you.
  14. P

    What should I assume when I see log?

    Today in a lecture in one of my engineering courses the professor said that mathematicians assume that when log is written the base is e. I had always learned to assume base 10, and simply write ln when I wanted the base as e. Is my professor correct or is that a bad habit to adapt?
  15. C

    Differential Equation Containing Natural Log of Negative e

    Hi I am working on a problem that ends up having the natural log of a negative e which I'm confused on how to find the explicit solution. The Problem: Find an explicit solution with C. y'-e^{-y}cos(x)=0 My Conclusion: First of all, I'm confused how I should solve this explicitly if I'm...
  16. C

    What is log (100)? Bachelor of Engineering? Civil Field?

    Without trying to lean anyone to either answer, I will post WHY I asked this question after I gather some data from the poll. thanks. Pretty simple question: What is log (100) ? If you can, please post what degree you have and what field, for example: Bachelor of Engineering: Civil Thanks.
  17. T

    Log problem (or by whatever method)

    Homework Statement a^2 + 2^a = 100, where a is an integer, find a. Homework Equations all laws of indices and laws of log, I think The Attempt at a Solution By trial and error, answer can be easily determined, which is 6. However, I am unsure how to approach this problem with...
  18. T

    Limiting Logarithm: x→0+ log(x-1+√(x^2+1))-logx

    Homework Statement limx→0+ log( x - 1 +\sqrt{x^2+1})-logx) Note that log is the same as loge same as ln. Couldn't find the three parallel lines symbol.The Attempt at a Solution I really don't know where to start with this one. Should I use the limit laws and bring the limit into the...
  19. F

    Convergence of Natural Log function with the limit comparison test

    Homework Statement Determine whether Ʃ(n from 1 to infinity) ln(n)/n^3 converges or diverges using the limit comparison test. Homework Equations I must use the limit comparison test to solve this problem-not allowed to use other tests. The Attempt at a Solution I know that the...
  20. J

    Maclaurin Series for Natural Log Function

    Homework Statement Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt(e) to four decimal places.Homework Equations The Attempt at a Solution \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + x^2/2 + x^3/6 + ... For this particular power series, I have: \sum_{n=0}^\infty...
  21. polygamma

    MHB Log Integral: $\displaystyle \frac{\pi^2}{b^2} \csc^2\left(\frac{\pi}{b}\right)$

    Show that for $ \displaystyle b>1, \ \int_{0}^{\infty} \frac{\ln x}{x^{b}-1} \ dx = \frac{\pi^{2}}{b^{2}} \csc^{2} \left(\frac{\pi}{b} \right)$.
  22. alyafey22

    MHB Prove Cubic Log Integral: -π^4/15

    Prove the following \int^1_0 \frac{\log^3(1-x)}{x}\, dx=-\frac{\pi^4}{15}
  23. P

    Natural Log of Product Solution

    Homework Statement L(θ) = ∏(θ/(2√xi)*e^(-θ√xi)),i=1, n Homework Equations The Attempt at a Solution -> θ2∏(1/(2√xi)*e^(-θ√xi)) taking natural log of both sides lnL(θ) = nlnθ + ln∏(1/(2√xi)*e^(-θ√xi)) = nlnθ + Ʃln(1/(2√xi)*e^(-θ√xi)) Ok so from what I understand the...
  24. Tyrannosaurus_

    Log. functions - comparing solutions, which is best?

    Homework Statement I'm not sure what my error is. Both solutions cannot be true.Homework Equations evaluating logarithms, power of logarithms?The Attempt at a Solution SOLUTION ONE =logxxn =n(logxx) =n(1) =nSOLUTION TWO =logxxn =(logxx)n =(1)n =1 Please help me understand what I've done...
  25. Albert1

    MHB Prove A < B with Log Inequality $\pi\approx3.1416$

    $\pi\approx3.1416$ $A=\dfrac{1}{log_5 19}+\dfrac{2}{log_3 19}+\dfrac{3}{log_2 19}$ $B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_3\pi}$ edit :$B=\dfrac{1}{log_2\pi}+\dfrac{1}{log_{\color{red}5}\pi}$ $Prove: \,\, A < B$
  26. alyafey22

    MHB Prove Log Gamma Integral: $\sqrt{2 \pi}$

    Prove the following \int^1_0 \ln\left( \Gamma (x) \right) \, dx = \ln \left( \sqrt{2 \pi } \right)
  27. D

    Useful Representations of Log Audio Frequency Spectrum

    Hi everyone, So I'm trying to basically generate a list of numbers between 20 and 20,000 (Hz) in log space that will give good resolution to parts of the audio spectrum that matter! After all that is the point of using log scale for frequency in the first place. The list generator I have...
  28. T

    Prove: f(z) = log z cannot be analytic

    I'm having difficulty completing the last problem of an assignment due tomorrow evening. I feel as if I'm missing something; every time I attempt the problem I get stuck or confused. "22. Use the conclusion of Exercise 21 and Example 13 of Section 6, Chapter 1, to prove that f(z) = log z...
  29. Y

    Exploring the Simplification of Logarithmic Expressions

    I want to verify this: 2ln(x)-ln(2x)=ln(x^2)-ln(2x)=ln\left(\frac{x^2}{2x}\right)=ln\left(\frac x 2\right) ln(2x)-ln(x)=\ln\left(\frac {2x}{x}\right)=ln(2) Thanks
  30. D

    How do I find derivatives involving natural logarithms and multiple variables?

    So I have an exam tomorrow, and the teacher provided a review. f(x) = ln(x + y) I remember that d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is 1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no...
  31. M

    MHB Understanding the derivative of natural log function

    Hope this is in the right place... I'm trying to understand why the derivative of ln(x) is 1/x while the derivative of something like ln(4) is 0. My knee-jerk reaction is to view 4 as representative of x, thereby giving me F'(x) ln(4) = 1/4, not 0. That would be the case, except ln(4) is a...
  32. E

    Integrating a natural log in double integral

    Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...
  33. B

    MHB Definite integral involving the natural log function

    I figured I would just add this new problem over here, rather than starting a new thread. Im looking to solve integration leading to arctan or arcsin results. \int_{1}^{e}\frac{3dx}{x(1+\ln(x)^2}) Looking at this, it feels like this has an arctan in the result, but I would have to multiply...
  34. K

    Radioactive decay - log question

    Homework Statement The amount (A) of cesium-137 remaining after t years is given by A=A_{0}*2^{\frac{-t}{30.3}} where A_{0} is the initial amount. In what year will the cesium-137 be 10% of that which was released at the Chernobyl disaster in 1986? Homework Equations...
  35. DeusAbscondus

    MHB Is This Use of Logarithmic Rules Correct in Solving for x?

    Would some kind soul please look over the following and check that use of the log rules, thought roundabout, is nonetheless correct? (thx kindly: I'm revising stuff I tried to cram last year) The set question: Solve for x: $$y=ln(x)+1$$ Answer given in text: $$y-1=ln(x)$$ $$\therefore \text...
  36. N

    Calculate the Tension and Angular Acceleration in Log After Release

    Homework Statement The uniform 69-lb log is supported by the two cables and used as a battering ram. If the log is released from rest in the position shown, calculate the initial tension induced in each cable immediately after release and the corresponding angular acceleration α of the cables...
  37. L

    Is this the correct way to solve a logarithmic equation?

    Homework Statement 2log2 x - log2 (x-3) = 2 The attempt at a solution So what I did was , expand the brackets 2log x - logx + log3 = 2 logx = 2-log3 logx = log 4 - log 3 log x = log ( 4-3) x= 4/3? Is this right?
  38. R

    Calculating dA/dB ratio for horizontal log?

    In the figure, a 105 kg uniform log hangs by two steel wires, A and B, both of radius 1.25 mm. Initially, wire A was 2.50 m long and 1.80 mm shorter than wire B. The log is now horizontal. Young's modulus for steel is 2.00 × 1011 N/m2. What is the ratio dA/dB? I'm honestly at a loss for how...
  39. E

    Log Function and Exponent Precedence

    Homework Statement This isn't really a specific problem, just a question if hand-writing log functions (or trig functions) is interpreted differently than when typing them into a calculator or something like Wolfram Alpha. Suppose you have this on paper: ln ex Is this the same as...
  40. K

    The Derivative of a log of a trig function

    Homework Statement I came across a question like this on a test today, and it says it would be helpful to simplify before differentiation. However, I could not find ways to simplify (1- cos(x))/(1+cos(x)). Homework Equations ln((1- cos (x))/(1+cos(x)))^7) The Attempt at a Solution...
  41. B

    Infinite Series with log natural Question

    Homework Statement ∞ Ʃ (-1)^(k+1) / kln(k) k=2 Homework Equations integral test, p test, comparison test, limit comparison, ratio test, root test. The Attempt at a Solution In class so far we have not learned the alternating series test so i can't use that test. So far I have...
  42. K

    Log Identity Proofs: Simplifying a^{log_{b}(c)}=c^{log_{b}(a)}

    Homework Statement a^{log_{b}(c)}=c^{log_{b}(a)} The Attempt at a Solution Take log_{a} of both sides: log_{a}(a^{log_{b}(c)})=log_{a}(c^{log_{b}(a)}) gives: log_{b}c=log_{b}alog_{a}c Looks like one more step for the RHS. I sort of see that the RHS should become log_{b}c and...
  43. A

    MHB What is the Derivative of f(x) = 3xlnx?

    if f(x)= 3x lnx, then f ' (x)=? i used f' ' (x)=3x(D lnx) + D (3x) (lnx) f ' '(x)=3x (1/x) + 3 (lnx) so... f ' '(x)=3+3lnx or 3(1+lnx). unfortunately this isn't one of the possible answers given. could one of you kind folks help me understand where i went wrong? thank you
  44. M

    EM algorithm convergence KF log likelihood decrease

    Hi everyone, Im running the KF to learn parameters of a model, the log likelihood of the p(Y_{k}|Y_{k-1}), however decreases. Can anyone advise, does this mean my implementation is wrong or can this just be the case. Advice appreciated Thanks
  45. P

    Derivative and integral of the natural log

    not really a problem, but more curious if we differentiate ln(2x) we get 2/(2x) = 1/x by the chain rule, but if we integrate 1/x we get ln|x|? Could anyone explain why this is the case, thanks.
  46. T

    Purpose of natural log in gibbs free energy equation

    Hi,Im just beginner and I m trying to learn integrals.I m just in starting phase,but still in few tences,not details...How or why we get logarithm in gibbs free energy equation?Because of integration of this equation or due to probability and statistics laws? Thanks
  47. M

    Is the rewritten form of ln(x2) valid?

    I'm reading back over a calculus book getting ready for an exam and I'm seeing a note that I don't understand. It says to make sure, when rewriting a ln function that the domain is the same, then it provides an example of when it's not the same, yet says nothing more. Is this rewritten form...
  48. S

    Log expansion for infinite solenoid

    Hello, I found an approximation for this log function: log \Bigg(\frac{\Lambda}{\rho} + \sqrt{1 + \frac{\Lambda^2}{\rho^2}} \Bigg), where \Lambda \rightarrow \infty . The above is approximated to the following, -log \bigg(\frac{\rho}{\rho_o} \bigg) + log \bigg(\frac{2 \Lambda}{\rho_o}...
  49. X

    Log base 2 is the same thing as square root?

    Hi, Is is correct to say that the logarithm of base 2 of a number x, is the same thing as the square root of a number x?
  50. G

    Renormalized vertex always a log?

    Is there a reason that one-loop corrections to vertex functions seem to always be logarithmic? If you write the vertex as a Taylor series in the external momenta, then the first couple of terms (say the constant and linear terms) diverge, but these divergence gets subtracted off, so you now...
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