What is Natural log: Definition and 147 Discussions

The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, loge x, or sometimes, if the base e is implicit, simply log x. Parentheses are sometimes added for clarity, giving ln(x), loge(x), or log(x). This is done particularly when the argument to the logarithm is not a single symbol, so as to prevent ambiguity.
The natural logarithm of x is the power to which e would have to be raised to equal x. For example, ln 7.5 is 2.0149..., because e2.0149... = 7.5. The natural logarithm of e itself, ln e, is 1, because e1 = e, while the natural logarithm of 1 is 0, since e0 = 1.
The natural logarithm can be defined for any positive real number a as the area under the curve y = 1/x from 1 to a (with the area being negative when 0 < a < 1). The simplicity of this definition, which is matched in many other formulas involving the natural logarithm, leads to the term "natural". The definition of the natural logarithm can then be extended to give logarithm values for negative numbers and for all non-zero complex numbers, although this leads to a multi-valued function: see Complex logarithm for more.
The natural logarithm function, if considered as a real-valued function of a real variable, is the inverse function of the exponential function, leading to the identities:











{\displaystyle {\begin{aligned}e^{\ln x}&=x\qquad {\text{if }}x>0,\\\ln e^{x}&=x.\end{aligned}}}
Like all logarithms, the natural logarithm maps multiplication of positive numbers into addition:





{\displaystyle \ln xy=\ln x+\ln y.}
Logarithms can be defined for any positive base other than 1, not only e. However, logarithms in other bases differ only by a constant multiplier from the natural logarithm, and can be defined in terms of the latter. For instance, the base-2 logarithm (also called the binary logarithm) is equal to the natural logarithm divided by ln 2, the natural logarithm of 2.
Logarithms are useful for solving equations in which the unknown appears as the exponent of some other quantity. For example, logarithms are used to solve for the half-life, decay constant, or unknown time in exponential decay problems. They are important in many branches of mathematics and scientific disciplines, and are used in finance to solve problems involving compound interest.

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  1. YouAreAwesome

    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...
  2. J

    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!
  3. R

    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.
  4. F

    Solving for time in an exponential equation

    Homework Statement In my book, there is a formula that gives the amount (in grams) of Radium in a jar after t years (100 grams were initially stored): R = 100⋅e-0.00043⋅t The book asks me to sketch the graph of the equation. I decided to find a point where the time elapsed equals the...
  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. 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?
  7. 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} +...
  8. 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...
  9. 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}...
  10. 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.
  11. T

    MHB Derivative of function with a natural log in the exponent

    Supposing we have $f(x) = {2}^{lnx}$, how would we find $f'(x)$?
  12. 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?
  13. 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 $- \ln\left({\frac{b}{a}}\right)$?
  14. 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 -...
  15. 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...
  16. S

    B Integrating velocity equation problem

    I've already completed the first question, but with number two, it's a different case. Here's my attempt: \frac { d{ v }_{ y } }{ dt } \quad =\quad -g\quad -\quad \beta { v }_{ y }\\ \frac { d{ v }_{ y } }{ -g\quad -\quad \beta { v }_{ y } } \quad =\quad dt\\ \int { \frac { d{ v }_{ y } }{...
  17. 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]} ?
  18. 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...
  19. A

    Problem with integration, relating natural log and arctanh

    Homework Statement The problem is actually from chemical kinetics, but my problem is solving the differential equation obtained. (dx(t))/(dt) = -j*x^2-k*x+k*a ; x is a function of t, and j,k,a are all real positive constants. Homework Equations I know this is a Ricatti type equation. But this...
  20. S

    Natural log differentiation question

    Homework Statement f(x) = 1/ln (10-x) -- I would assume it to be a fairly simple equation, but I am screwing it upHomework Equations What is f'(x)?The Attempt at a Solution f'(x) = (ln (10-x))^-1 = -(ln (10-x))^-2 * -1 * 1/(10-x) -- 2 negatives cancel out = 1/(10-x) (ln(10-x))2 --...
  21. S

    Derivative of natural log function questions

    Homework Statement Hello, I had a few derivative of the natural logarithm functions questions. It seems like it should be fairly straightforward, but I am turning it into a pig’s ear.On my honor, none of these are problems on an assessment per se, however, they are not materially different than...
  22. T

    Derivative of Natural Log: How to Solve Number 3 on Homework Assignment

    Homework Statement I posted a picture of it and my attempt it is number 3 Homework EquationsThe Attempt at a Solution I tried using log properties and I am not sure what went wrong and how to arrive at the correct answer. Mod note: Messy, disorganized image deleted.
  23. C

    Muon Decay: Homework Statement & Solution

    Homework Statement By taking the natural logarithms of each side, that this equation can be linearized by making a semi logarithmic plot. Identify the variables and state what the slope , and y-intercept would represent. Homework Equations F=Fo*e^-(d/cτ) The Attempt at a Solution I tried...
  24. lovexmango

    What is the parametrization of the graph of ln(x)?

    Let k(x) be the curvature of y=ln(x) at x. Find the limit as x approaches to the positive infinity of k(x). At what point does the curve have maximum curvature? You're supposed to parametrize the graph of ln(x), which I found to be x(t)=(t,ln(t)). And you're not allowed to use the formula with...
  25. M

    Properties of natural log and absolute value

    Homework Statement from -lnIxI to lnI x^-1I , I try to go from -lnIxI to lnI x^-1I by using some properties. Homework Equations - lnIxI The Attempt at a Solution First I write the -lnIxI as -1*lnIxI and then I use -1 as an exponent to absolute value of x in the natural log that is ln (...
  26. D

    MHB Natural log of a complex number.

    Evaluate the following logarithms, expressing the answers in rectangular form a. $\ln1$, $Ln1$ b. $\ln(3-j4)$, $Ln(3-j4)$ I know that the log of a complex number z is given as $\ln z=\ln|z|+argz$ but I still don't know how to use this fact to solve the problems above. I'm having a hard...
  27. B

    Exploring the Inverse Relationship between e^-1 and Natural Log e

    Why is e^-1 the inverse of natural log e? Thank you
  28. sheldonrocks97

    Why is natural log abbreviated as ln and not nl ?

    Why is natural log abbreviated as "ln" and not "nl"? I've been taking calculus for a while now and I was just wondering why natural logarithm is abbreviated as "ln" and not "nl". I'm just curious!
  29. 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...
  30. 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(...
  31. 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...
  32. 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...
  33. 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...
  34. 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...
  35. 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
  36. 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...
  37. 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...
  38. 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...
  39. 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...
  40. 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
  41. 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.
  42. 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
  43. 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...
  44. D

    Exploring the Relationship between -ln(-∞) and ln(∞)

    For instance, say I have -ln(-∞) Does the negative sign on the natural log cancel with the negative sign on the infinity? Is this true? -ln(-∞) = ln(∞) Thank you -Drc
  45. J

    Natural Log on Radioactive Decay Formula

    Homework Statement The initial amount of radioactive atoms on a sample of 24Na is 10^10. It's half-life corresponds to 15 hours. Give the amount of 24Na atoms that will disintegrate in 1 day.Homework Equations I started to solve it using the formula N=Initial Amount of Atoms /...
  46. Q

    Why do we use the natural log in the derivative of an exponential function?

    I recently struck a question that I have not been able to find an answer to. I feel like I'm missing something obvious, so I've come here for help. The derivative of a^{x} is a^{x}lna. The explanation that Stewart 5e gives is: \frac{d}{dx}a^{x} = \frac{d}{dx}e^{(lna)x} =...
  47. H

    Natural Log : seems as a discontinous function

    The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as lim_{x\rightarrow c} f(x) = f(c) from the positive and negative sides . For ln(x) (the natural log...
  48. S

    Natural log (ln x) derivative question

    Homework Statement This is to help out a 40something calc student -- thank you all in advance for your help Homework Equations If f (y) = ln ln ln x, what is ∂y/∂x? The Attempt at a Solution I came up with 1/x, which I got by applying ∂y/∂x ln x = 1/x three times, is this...