# What is Log: Definition and 632 Discussions

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. ### B Creating Log Tables: Briggs' 400-Year-Old Accomplishment

I haven't used a log table in a long time and the one I used in High School went from 10 to 99 with four significant figures for the logarithms. I wondered how did Briggs create a log table 400 years ago? How did he find the logarithm of 7(base 10)? I looked around the internet and You Tube and...
2. ### What Is the Next Step in Solving This Logarithmic Equation?

I started off by using law of logs to divide the logb (6x/18) but i dont know what to do after, please help.
3. ### Milestones of a human lifetime on a log scale

In another thread the concept of how many seconds in a year came up and it made me think of putting milestones of a human lifetime on a log scale. Units are seconds. Please feel free to amend or change or just make it better. Maybe an actual plot? What strikes me is most of life is in the last...
4. ### POTW What is the Log Concavity of Determinants for Positive Definite Matrices?

Let ##A, B\in M_n(\mathbb{R})## be positive definite matrices. Prove that for ##0\le t\le 1##, $$\det(tA + (1 - t)B) \ge \det(A)^t\det(B)^{1-t}$$
5. ### B Can a log have multiple bases?

Hi, I tutor maths to High School students. I had a question today that I was unsure of. Can the natural log be to the base 2? The student brought the question to me from their maths exam where the question was: Differentiate ln(base2) x^2 If the natural log is the inverse of e then how does...
6. ### Mathematica How to simplify expressions in Mathematica with log and exp in them?

Hi, This is a pretty simple question, but I am new to Mathematica so I am not sure if I am missing something obvious. Question: How do I make the expression ## e^{log(z)} ## return z? Attempt: I have used all of the following combinations and all of them return ## e^{log(z)}##. Are there any...
7. ### Appliances Reducing output of log burning stove

My log burner is just too powerful for everyday use. It is spec'd at 5kW and has an area of about 200X400mm. It's a single fuel design with no bottom vent. The air control is quite clever. It has an 'air curtain' supply which comes down from the top and two adjustable slots in the doors to get...
8. ### MHB How can I solve the problem of $e^x-\ln{x}=4$ without a calculator?

solve $e^x-\ln{x}=4\quad x\approx1.48 \quad x\approx 005$ ok I could only do this with a calculator but need steps
9. ### Limit calculation involving log and trig functions

This was the question, The above solution is the one that I got originally by the question setters, Below are my attempts (I don't know why is the size of image automatically reduced but hope that its clear enough to understand), As you can see that both these methods give different answers...
10. ### My wife's computer won't log on to the bank account

My wife cannot log onto Wells Fargo account. She talked to the WF assistants, they directed her to delete cookies and history. Still doesn't work. She tried on my computer, it works. So problem is on her computer. She uses FireFox, I went in, deleted ALL history( thousands of them), I deleted...
11. ### MHB Log likelihood and Maximum likelihood

I'm not sure how to get this first derivative (mainly where does the 4 come from?) I know x̄ is the sample mean (which I think is 1/2?) Can someone suggest where to start with finding the log-likelihood? I know the mass function of a binomial distribution is: Thanks!
12. ### Kinematics Question: A motorboat and a log in a stream

I found vlog=1.9/t and vb-v=1.15/t and vb+v=d+1.15/60.8 don't know what to do now
13. ### I Can I Use a Natural Log Function for Least Square Fitting?

Hello, I'm trying to follow Wolfram to do a least square fitting. There are multiple summations in the two equations to find the coefficients. Are the i's the same in this case? Thanks!
14. ### Stuck doing parametric natural log graphs

I got stuck when i eliminated the parameter.
15. ### B I do not understand Log tables

I was watching this video about how they invented log tables and some things I do not understand. At 5:12 he changes the scaling from 1's to 10's I don't understand how this is allowed. The number represents the number of times the base was multiplied by itself. In the seconds before this...
16. ### Understanding Iterative Derivatives of Log Functions

I have never used induction to justify the derivative to a function, so I don't know where to start. Does anyone have some tips?
17. ### What is the optimal communication time for a given set of parameters?

Hi, I am trying to solve the following equation: ##\mathbf (t_s + t_wm)log(p)+ 2t_h (\sqrt{p} - 1) ## ##\mathbf (10+0.01 * 1000) log(36) + 2 * 2(5) ## I think log(36) would be 18. Note log is ##\mathbf log_2 ## Somebody please guide me. Zulfi.
18. ### I What went wrong in the algebraic steps for ln rules?

Given ln(ab) = b⋅ln(a) Then ln(1x) = x⋅ln(1) Also ln(2x) = x⋅ln(2) Say ln(2x) = ln(1x) Then Also x⋅ln(2) = x⋅ln(1) But, dividing both sides by x ln(2) ≠ ln(1) Similarly, x⋅ln(2) = x⋅ln(1) Dividing both sides by x and ln(2) 1 ≠ 0 But we know x = 0 as per the original statement...
19. ### 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...
20. ### I Proof of Log Convexity Theorem & Its Derivation

Hi, in this link https://math.stackexchange.com/questions/2605752/proof-log-convex-implies-convexity , there is a theorem that labelled with yellow. I do not know the name of theorem and would like to know it's derivation. Could you please help me?

Homework Statement "Find ##\int_0^x \frac{6\ dt}{2t+1}##." Homework Equations ##\int \frac{\ dx}{x} = \ ln(x)## for ##x>0## The Attempt at a Solution Method 1: Let ##u=2t+1##, ##\frac{1}{2} du=\ dt##. ##\int_0^x \frac{6\ dt}{2t+1}=3\int_1^{2t+1} \frac{du}{u}=3\ ln(u)|_1^{2t+1}=3\ ln(2t+1)##...
22. ### B Triangle of Powers: Revolutionary Math Notation

Recently I saw this clip on YouTube which has a revolutionary idea of mathematical notations,What do you think about it?
23. ### Statistical Mechanics- moments/cumulants, log expansion

Homework Statement Using log taylor expansion to express cumulants in terms of moments I have worked through the expansion- ##log(1+\epsilon)= ...## see thumbnail- and that's ok; my question is why does the expansion hold, i.e. all i can see is it must be that ##k## is small- how is this...
24. ### Convergence of Series Involving Logarithms and Reciprocals

Homework Statement Show that ##\sum_{n=1}^{\infty}\frac{\log (1+1/n)}{n}## converges. Homework EquationsThe Attempt at a Solution If I take for granted the inequality ##\log (1+1/n) < 1/n##, I can easily show that this converges. My problem is is that I am not seeing how to prove convergence...
25. ### I Principal branch of the log function

I'm learning complex analysis right now, and I'm reading from Joseph Taylor's Complex Variables. On Theorem 1.4.8, it says "If a log is the branch of the log function determined by an interval I, then log agrees with the ordinary natural log function on the positive real numbers if and only if...
26. ### MHB Finding Domain for Natural Log with Exponent f(x)=ln(x^2−5x)

I just asked a similar question, but I got help for that one, and now I am stumped again. I need to find the domain for f(x) = ln(x^2-5x) What's confusing me is how to deal with the exponent. I can't think of a way to get around it.
27. ### Crashed Android on ZTE Phone -- How to read Log files

Hi, My Android ZTE phone is not loading the OS, not completing POST andgiving me an Android Recovery error message . How can I read/use log files to recover phone? Here is the error message, and the files listed in it, which I can access using on/volume keys (On key selects, Volume allows me...
28. ### MHB Plus or minus question for the complex log

Dear Everybody, I am having troubles figuring out why the plus or minus sign in this problem. The question is: Solve the equation $\sin\left({z}\right)=2$ for $z$ by using $\arcsin\left({z}\right)$ The work for this problem is the following: $\sin\left({z}\right)=2$...
29. ### Binary Representation: Log or Ln?

What is the difference between log and ln?? [Thread moved to General Math forum by moderator]
30. ### I How to derive this log related integration formula?

here is a link with the formula: https://portal.uea.ac.uk/documents/6207125/8199714/steps+into+calculus+integration+and+natural+logarithms.pdf i'm talking about the formula that says the integral of f'(x)/f(x)dx = ln(f(x))+C it's kind of hard to put this into Google. where does this formula...
31. ### Maximization problem — Stiffest beam that can be cut from a log

Homework Statement Homework Equations Pitagora's: ##~a^2+b^2=c^2## Maxima/minima are where the first derivative is 0 The Attempt at a Solution $$\left( \frac{a}{2} \right)^2+\left( \frac{b}{2} \right)^2=r^2~\rightarrow~b^2=\frac{16r^2}{a^2}$$ The strength S has the proportion coefficient k...
32. ### Analyzing the Oscillation of a Plank on a Log

Homework Statement Homework EquationsThe Attempt at a Solution C is the point of contact and G is the CM of the plank . x be the distance between G and C .Since plank always remain in contact , x=aθ When the line joining the point of contact C with the center makes an angle θ with the...
33. ### MHB What is the difference between log and ln?

In simple words, what is the difference between log and ln?
34. ### Designing a Log Cutting Mechanism - Get Suggestions Here

Good afternoon guys, I want to discuss with you guys a few things... So i already design a log cutting mechanism as you can see in the picture below. But there is so much defect on my mechanism.. The biggest thing that stressed me out, its still uneasy to use because after cut the log, the user...
35. ### How to calculate the force to cut a log?

Good morning guys, So i want to make mechanism to cut log with diameter 5 cm and 120 cm length. So how i can calculate the force to cut this log perpendicular and paralel. What is tbe formula i can use? And how i can determine the saw thickness and rpm needed to cut this log? Thanks guys for...
36. ### B Solve for x (natural log equation)

I just wanted to double check that I was doing this right ( if the math in my process makes sense). Question: Solve for x given the following equation. eex= 2 I found the ln of both sides, so : ex= ln2 and then found ln again x= ln(ln(2)) Was this the correct way of doing this? I...
37. ### I Calculation of x from log equation

Hi I have the following equation: Log(x/y)=A*(e^z(-1)) I know values of y, A, z. I need to calculate x from this equation. I need to ‘extract’ x first therefore Log(x)-Log(y)= A*(e^z(-1)) Log(x) = A*(e^z(-1)) +Log(y) x=10^[ A*(e^z(-1)) +Log(y)] I am not sure whether my formula modification...
38. ### MHB Positive Values of $a$, $b$, $c$, and $d$ for Log Inequality

Determine the positive values of $a,\,b,\,c$ and $d$ such that $\log_{d+3} (a + 3)>\log_{c+2} (d + 2)>\log_{b+1} (c + 1) >\log_a (b)$.

Homework Statement I'm having a hard time differentiating when to use log instead of ln, vice versa. Are there any general rules to follow? For example I have to evaluate 4u^-3 + u^-1. Homework Equations f'(1/u) = log u f'(1/u) = ln u The Attempt at a Solution I put -2u^-2 + log(u) but the...
40. ### Log inside log -- find the x in its max domains?

Homework Statement Homework Equations log_2 x = y 2^y = x 3^2^y The Attempt at a Solution log_2 x = y 2^y = x log_2 {log _3 {log _2 { log_3 {2^y} } } } what am I suppose to do?
41. ### MHB Visual proof log(ab) = log a + log b

Hi, I'm looking for a visual proof log(ab) = log a + log b I've seen diagrams where the values are measured out, but it's not immediately obvious why this holds. Is there an intuitive way to illustrate this? Also, are there other functions with this property.
42. ### MHB How to Solve a Log Equation with Different Bases?

Solve the log equation log_16 (3x-1)=log_4 (3x) + log_4 (0.5) Can someone get me started? The left side has base 16 and the right side base 4. How is this done when two different bases are involved?
43. ### I Integral involving square and log

I have this integral that when solved, involves squares and natural logs, where ##A\,##,##\,b\,##, and ##\,x_e\,## are constants while ##x## is a variable. ##A = \int_{x_e}^{x} \frac{x^2 - b^2}{x} dx = \int_{x_e}^{x} x \, dx -b^2 \int_{x_e}^{x} \frac{dx}{x} = \frac{x^2}{2} - \frac{x_e^2}{2} -...
44. ### MHB If y=sin5 + log base 10 x+ 2 sec x find dy/dx

If y=sin5 + log base 10 x + 2 sec x find dy/dx Please solve the problem
45. ### I How Does Log(z) Expand at Infinity?

Hi, see attached PdF file for my question concerning serie expansion of log(z) at infinity. Thank you Belgium 12
46. ### How Do We Simplify and Interpret the Expression e^(ln(4)/2)?

Homework Statement $$ln(x^2)=4$$ Homework Equations ##a^{log_a(x)}=x## ##log_a(a^x)=x## The Attempt at a Solutionln(x^2)=4 <=> ##2ln(x)=ln(4)## <=> ##ln(x)= [ln(4)]/2## <=> ##log_e(x)= [ln(4)]/2## <=> ##e^{ln(4)/2}=x## <=> ##[e^{ln(4)}]^{1/2}## <=> ##sqrt(e^{ln(4)})##...
47. ### I Free or cheap app for log plots

Trying to complete some homework help I got a bit stuck on an elementary or common problem that was always a bit difficult. I have a function f(x) and just want to plot log f(x) against log x. This is common in chemistry and biochemistry. It should have been easy. :redface: I have picked up...
48. ### Why (a log x = b log x / b log a)?

Could someone please explain to me why this equation holds: alog x = blog x / blog a For all the all other rules concerning logarithms I could derive them from the rules for exponentiation, except for this rule. Could someone please explain to me how this derivation works? And how the value of...
49. ### Bi-geometrical mean using logs -- don't get the same result

Homework Statement Hello! I am trying to compute the bi-geometrical mean on data that contains negatives. But before that I wanted the test the formula that accounts only for positive values using the sum of their logarithms. By doing so I don't get the result I compute by using the "usual" geo...
50. ### MHB Understanding the Horizontal Shift in Logarithmic Functions

If we have log 3 (x+9) + 2 = y. It states that we have a HT 9 units left and 2 units up. But if we manipulate that too log 3 (x+9) = y - 2 And convert to Exp form. This gives me 3^(y-2) = x + 9 or 3^(y-2) - 9 = x This looks to me more like a HT 2 units right and a vt 9 units down. I don't...