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. G

    B Log(x), an easy and useful way to calculate it

    ½Logb(x2-1)≈logb(x) This is an easy and useful way to calculate the log of any natural number, including primes, it won't ever give a precise result, obviously (because of the -1), but as "x2-1" will always have divisors smaller than "x", you can easily calculate the approximation by using the...
  2. 1

    I Rounding decimals to ruler scale (log, pow math?)

    I have data (all less than 1) and I need to round down to the nearest 1/2, 1/4, 1/8, 1/16, and 1/32. I have no idea how to do this mathematically, but I'm guessing I use log or pow functions. For example: 0.81 = 0.5 or 1/2 0.33 = 0.25 or 1/4 0.18 = 0.125 or 1/8 Any help would be appreciated...
  3. A

    Fortran How to change data from linear scale to log scale

    I am trying to calculate and draw a relation between (IL) and the Growth rate (m). I have used Il as range from 10^15 to 10^18 in the calculation and when I draw the relation (x-axis ) is Il and (y-axis ) is m. I converted the IL to be in log scale. I noticed unacceptable output. So, I am...
  4. R

    Mastering Logarithms: Simplifying Complex Expressions with Multiple Logs

    Homework Statement can we put 3log2(x)-4log(y)+log2(5) in one logarithm it try in all the ways but i can't find the solution . Homework Equations loga(b)=logx(b)/logx(a) log(b*a)=log(b)+log(a) The Attempt at a Solution log2(5x^3)-log(y^4) log2(5x^3)-log2(y^4)/log2(10)
  5. N

    Proving Natural Log Proof: ln|1+σx|

    Homework Statement Prove the following statement: ln|1+\sigma x | = \frac{1}{2} ln|1-x^2| + \frac{\sigma}{2} ln| \frac{ |1+x|}{|1-x|} Homework EquationsThe Attempt at a Solution Starting from right to left would be easier: = \frac{1}{2} ln|(1+x)(1-x)| + \frac{\sigma}{2} ln| 1+x| -...
  6. N

    Solving for n: power and log rules refresh

    Homework Statement A computer can perform 1010 ops/s. Assume 1 op per 1 input. Given the following algorithmic complexities how many inputs can be performed in an hour. n2 n3 100n2 n log n 2n 22n Homework EquationsThe Attempt at a Solution [/B] C = 1·1010 ops/s · 3.6·102 s/hr C = 3.6·1012...
  7. A

    Inifinity limit with natural log

    Homework Statement Limx--> ∞ Ln(x^2-1) -Ln(2x^2+3) Homework EquationsThe Attempt at a Solution Ln(x^2-1)/(2x^2+3) Then I divided the top and bottom by x^2 so in the end I got (1/2). Is this right?
  8. K

    MHB Test for Cauchy sequence (with limsup and log)

    If $\{x_n\}_{n \ge 1}$ is real sequence and $\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n|<0$, prove that $\{x_n\}$ is Cauchy sequence. My work: Let $a=\limsup\limits_{n \to \infty} \frac{1}{n} \log |x_{n+1}-x_n| <0$. Then, for every $\varepsilon >0$ there exist $N \in...
  9. F

    A Anyone can have the idea how to solve the log of a power ser

    hi, anyone can have the idea how to solve the log of a power series? i.e. log(1+ Σ_{j=1}^{∞}(1/z^{2 j-1}) f^{j})
  10. D

    MHB How Do You Solve These Complex Logarithmic and Polynomial Equations?

    First Question: Solve the following system of equations log{x+1}y=2 log{y+1}x=1/4 Work: Turned them into equations (x+1)^2=y (y+1)^(1/4)=x Substituted second equation into the first equation ((y+1)^(1/4)+1)^2=y factored out and eventually got ((y+1)^1/4)^2+2((y+1)^1/4)+1=y Tried...
  11. M

    I A question about the log of a rational function

    We have the rational function : $$f(x)=\frac{(1+ix)^{n}-1}{(1-ix)^{n}-1}\left(\frac{1-ix}{1+ix}\right)^{n/2}\;\;\;,\;\;n\in \mathbb{Z}^{+}$$ It's not hard to prove that ...
  12. perplexabot

    A Derivative of log of normal distribution

    Hey all, I've had this point of confusion for a bit and I have thought that with time I may be able to clear it out myself. Nope, hasn't happened. I think I need help. Let us say we have the following \phi_{k+1}=\phi_{k}+v_k where, v_k\overset{iid}{\sim}\mathcal{N}(0,\sigma^2) and...
  13. T

    MHB Finding Maclaurin series of a natural log function

    I need to find the Maclaurin series of this function: $$f(x) = ln(1 - x^2)$$ I know that $ln(1 + x)$ equals $$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1} x^n}{n}$$ Or, $x - \frac{x^2}{2} + \frac{x^3}{3} ...$ If I swap in $-x^2$ for x, I get: $$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...
  14. T

    MHB Maclaurin series for natural log function

    I'm examining the Maclaurin series for $f(x) = ln(x + 1)$. It is fairly straightforward but there are a few details I'm not getting. So: $$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$ which equals: $A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$ I'm...
  15. karush

    MHB S6.7.1.13 natural log Integration

    $\large {S6.7.1.13}$ $\tiny\text {natural log Integration}$ $$\displaystyle \int e^{\sqrt[3]{x}} \, dx = 3\left(x^\frac{2}{3} -2\sqrt[3]{x} +2\right){e}^\sqrt[3]{x}+C \\ u=x^{1/3} \therefore 3{x}^{\frac{2}{3}} du = dx $$ $\text{not sure if this is how to start to get to a 3 term answer}...
  16. T

    MHB What is the Integral of 1 over 2 to the Natural Log of X?

    I have this integral $$\int_{}^{}\frac{1}{{2}^{lnx}} \,dx$$ I'm not sure the best way to do it. I tried u-substitution: $u = {2}^{lnx}$ and thus $u = {x}^{ln2}$, therefore $du = ln2({n}^{ln2 - 1}) dx$. However, not sure how to proceed from there.
  17. T

    MHB Derivative of function with a natural log in the exponent

    Supposing we have $f(x) = {2}^{lnx}$, how would we find $f'(x)$?
  18. T

    MHB Natural Log Inequality: True or Misunderstanding?

    I was talking to my professor and she said that $(ln n)^a < n$ for all values of $a$. Is this true or was I misunderstanding?
  19. karush

    MHB S6-7.1.79 log integral u substitution

    $\large{7.R.79} $ $\tiny\text{UHW 242 log integral }$ https://www.physicsforums.com/attachments/5717 $$\begin{align} \displaystyle x& = \frac{1}{u} & {u}^{2 }du&={dx } \end{align} $$ $$I=\int_{0}^{\infty} \frac {\ln\left({\frac{1}{u}}\right)} {1+\frac{1}{{u}^{2 }}} {u}^{2} \,du \\ Stuck🐮...
  20. T

    MHB Natural Log Rule: $\frac{a}{b}=-\frac{b}{a}$?

    If I have $\ln\left({a}\right) - \ln\left({b}\right)$ that would equal $\ln\left({\frac{a}{b}}\right)$ or $-(\ln\left({b}\right) - \ln\left({a}\right))$ which is also $- \ln\left({\frac{b}{a}}\right)$. So does this mean $\ln\left({\frac{a}{b}}\right)$ equals $- \ln\left({\frac{b}{a}}\right)$?
  21. T

    MHB Limit of Natural Log Sequence: How to Find It Using L'Hopital's Rule?

    I have this sequence: $${a}_{n} = \ln \left(\frac{12n + 2}{-9 + 4n}\right)$$ I need to find the limit of this sequence. How can I go about this? Do I need to apply L'Hopitals rule? I'm unsure how to simplify this expression. If I use the rule $\ln(\frac{a}{b}) = \ln a - \ln b$ I get $\infty -...
  22. karush

    MHB What is the integral of $xe^{2x}$ divided by the square of $1+2x$?

    $\tiny\text{LCC 206 {7.r.39} Integral log }$ $$\int\frac{xe^{2x}}{\left(1+2x\right)^2 }\ dx $$ Since $2x$ is in numerator and denominator thot it might be A good candidate for $\begin{align}\displaystyle u& = 2x & du&= 2 \ d{t} \\ \end{align}$ Then...
  23. karush

    MHB How Can We Integrate This Complex Logarithmic Function?

    $\tiny\text{LCC 206 {7.R.31} Integral log }$ $$\displaystyle I=\int_0^{\ln\left({10}\right)} \frac{e^{x}\sqrt{e^{x}-1}}{e^{x}+8} \ dx \\ \begin{align}\displaystyle u& = e^{x}-1 & du&= e^{x} \ d{x} \\ \end{align} \\ I=\int_0^{\ln\left({10}\right)} \frac{\sqrt{u}}{u+9} \ du $$...
  24. Jezza

    How Can the Limit of (ln(1+x))^x as x Approaches 0 Be Evaluated Correctly?

    Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...
  25. K

    Bullets stuck in a log and speed it up

    Homework Statement A cannon shoots bullets of mass m0 and velocity v0 at a massive log of wood of mass 100m0. the interval between the bullets is t0. the surface is smooth. the bullets are stuck in the wood. the penetration time of the bullets is very short, relative to t0. 1) What's the log's...
  26. Evangeline101

    Forces 2 dimensions: How far have the tugboats moved the log

    Homework Statement Two tugboats are pulling on a large log, as shown in the following diagram. The log has a mass of 250 kg and is initially at rest. How far have the tugboats moved the log after 10 s? Homework Equations c2 = a2 + b2 - 2abcosC c = (a2 + b2- 2abcosC)1/2 The Attempt at a...
  27. C

    How to Solve a Logarithmic Equation on a Given Base?

    Homework Statement solve the log x²-4x-5 on the base x-2.[/B] Homework Equations I read a comment from someone that also saw the question that says: The Attempt at a Solution If it was x²-4x+4, that would be igual (x-2)² and the answer would be 2, but I have no idea the guy got these...
  28. M

    MHB Can Logarithmic Functions Be Expressed as Infinite Series?

    Hi, I'm stuck on the following proof: \log[3] = \frac 1{729} \sum_{k=0}^\infty \frac 1{729^k} \left[\frac{729}{6k+1}+\frac{81}{6k+2}+\frac{81}{6k+3}+\frac 9{6k+4}+\frac 9{6k+5}+\frac 1{6k+6}\right] Manipulating and converting summands to integrals of the form $x^{-(6k+n)}$ over {x,0,3} seems...
  29. J

    How Do You Solve Exponential Equations with Different Bases?

    Homework Statement (1/25)^x+3=125^x+3 Homework Equations I have used log on both sides but keep getting an incorrect answer. The Attempt at a Solution How would solve this type of equation. I set the bases both to 5, make them equal to each other and i can't get the right answer.
  30. P

    Simplifying a log expression with identities

    I was supposed to simplify the expression ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## and apparently it’s wrong. Where’s the mistake? Is it not simplified enough or . . . ? ##\ln |\cot {x}|+\ln |\tan {x}\cos {x}|## ##=\ln |\frac {\cos {x}}{\sin {x}}|+\ln |\frac {\sin {x}}{\cos {x}}\cdot \cos...
  31. K

    MHB Lagrangian with log and summation

    This is a microeconomics problem that I am trying to solve. I am uncertain whether my FOCs are correct. Thank you. The objective function: ui(x1i, x2i….xLi) = Σllog[xli]; The constraint: ΣLl=1p1xl ≤ w L: Σllog[xli] + λ (w - ΣLl=1p1xl) FOCs are: L1 = 1/x1 – λ(w-p1) =0 L2 = 1/x2 – λ(w-p2)...
  32. S

    Proving the Variance of the Logarithmic Series Distribution

    Homework Statement \textbf{The Logarithmic Series Distribution}. We will examine the properties of a the Logarithmic Series Distribution. We will check that is is a probability function and compute a general term for factorial moments and, hence, compute its mean and variance. This distribution...
  33. G

    Easy and useful way to calculate Log(a+b)

    a>b ⇒ Lim(a–b)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) b>a ⇒ Lim(b–a)→0 Logc(a+b) ≈ Logc(a) + Logc(√a⋅b) a∈ℝ b∈ℝ c∈ℝ / c>0
  34. G

    I I found a *useful* method to calculate log(a+b), check it out

    So, I found this method, I don't think I was the first to, though, but I don't see any post related to this anywhere on the internet, so maybe there's a slim chance I was the first? Anyway, it doesn't really matter. The method does not give the precise result, only approximations, but I find it...
  35. R

    Why does a log float horizontally?

    Homework Statement Why does a log float horizontally as opposed to vertically? Homework Equations Nah The Attempt at a Solution I can kind of imagine why a vertical log would be in unstable equilibrium, but the thought isn't quite as logically cohesive as I'm comfortable with, so could...
  36. J

    Run-time error M6201: math -log: DOMAIN error

    Hi all, I am facing the problem while executing my fortran (77) program. please help me how to fix it. As I understood, the error means that the domain of log function has included zero values or negative values...But I could not understand...In my case... there are no zero or negative...
  37. S

    Understanding the Manipulation of Natural Logarithms

    How is it true that: Log[L+(Z^2+L^2)^(1/2)] - Log[-L+(Z^2+L^2)^(1/2)] = 2{Log[L+(Z^2+L^2)^(1/2)] - Log[Z]} ?
  38. B

    Log Derivative of a Function: Find Derivatives of f(c,l) with Example"

    Homework Statement f(c,l) = log(c - ψ(1-l)^θ ) What is the derivative of this function wrt. l and c?Homework Equations I know that the derivative of log (x) = 1/x The Attempt at a Solution I got wrt c: 1/ c - ψ(1-l)θ and wrt l: θψ(1-l)^θ-1 / c - ψ(1-l)^θ
  39. E

    Is My Solution to This Log Question Correct?

    Hi folks, I'm revisiting logs for the first time in a long time through distance education and I was wondering if someone could have a look over a question I've answered and let me know if I've done it correctly or if I'm way off please Find x if Log3(10x – 1) – 2 = 2log3x I instantly divide...
  40. NATURE.M

    Derivative of Log Likelihood Function

    So looking through my notes I can't seem to understand how to get from one step to the next. I have attached a screenshot of the 2 lines I'm very confused about. Thanks. BTW: The equations are for the log likelihood in a mixture of gaussians model EDIT: To elaborate I am particularly...
  41. D

    Normalisation of Experimental Data

    Hi all, I am currently operating a piece of equipment that essentially collects particles and separates them based on their size. Essentially you have 8 stages, and each stage has a differing size of particles it collects. For example: Stage----Size of Particles (D) (um)-----Mass Collected (M)...
  42. S

    Simplifying natural log of complex number

    Homework Statement The problem is to sketch lines of constant u and v in the image plane for the function Log[(z+1)/(z-1)]. Homework Equations z=x+iy The Attempt at a Solution In order to do this I have to get the expression into u+iv form, so that I can read off and manipulate the u and v...
  43. brainbaby

    Operation of transistor in Opamp log amplifier

    In the figure it could also be possible to connect the transistor Q1 as a diode by shorting its collector with its base ..but instead it is done by keeping both base and collector at ground... The text have to say that if it would be connected the other way(i.e by shorting collector and base...
  44. D

    Binary search math, log base 2 (n) issues

    Homework Statement Hello! Binary search requires O( log2(n) ) operations in the worst case. My math doesn't add up. I would be grateful for help. Homework Equations For example: Find number 8 in the set 1234578. The Attempt at a Solution The set contains n = 7 elements. 1) take the middle...
  45. B

    Integrals log x in terms of dilogarithm

    Hi members,see attached PdF file.Have you any idea to prove thes integrals Thank you
  46. J

    Checking for Accuracy in homework

    I noticed the scan was cut off on the second image at the bottom right, but I came up with x= 31/5 My first test in Calc I begins tomorrow and I want to know that I'm headed in the right direction. I think I understand to some extent how logarithms can be expanded and condensed though I'm...
  47. D

    Finite field with hard discrete log for both groups

    If there a finite field where both group structures have hard discrete logs? Discrete log in the additive group means multiplicative inverse.
  48. 22990atinesh

    How to prove that (log logn)×(log log log n) = Ω(logn)

    Is ##\log \log n \times \log \log \log n = \Omega(\log n)## How can we prove it. Actually I'm trying to prove that ##f(n) = \lceil(\log \log n)\rceil !## is polynomially bounded. It means ##c_1 n^{k_1} \leq f(n) \leq c_2 n^{k_2} \quad \forall n > n_0## ##m_1 \log n \leq \log [f(n)] \leq m_2...
  49. C

    MHB Change of bases with log tables

    Hi everyone, I need some help to solve this problem: The direction states to find the value by using the log table $\log_{3}\left({825.6}\right)$ Work: I using the change of base: $\log_{3}\left({825.6}\right)=\frac{\log\left({825.6}\right)}{\log\left({3}\right)}$ I look up the values of...
  50. J-dizzal

    Log hanging from 2 steel wires. stress/strain youngs modulus

    Homework Statement In the figure, a 128 kg uniform log hangs by two steel wires, A and B, both of radius 1.15 mm. Initially, wire A was 2.80 m long and 2.00 mm shorter than wire B. The log is now horizontal. Young's modulus for steel is 2.00 × 1011 N/m2. What are the magnitudes of the forces...
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