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:
e
ln
x
=
x
if
x
>
0
,
ln
e
x
=
x
.
{\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:
ln
x
y
=
ln
x
+
ln
y
.
{\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.
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...
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!
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.
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...
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| -...
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?
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:
$$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...
I'm examining the Maclaurin series for $f(x) = ln(x + 1)$.
It is fairly straightforward but there are a few details I'm not getting.
So:
$$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$
which equals:
$A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$
I'm...
$\large {S6.7.1.13}$
$\tiny\text {natural log Integration}$
$$\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 $$
$\text{not sure if this is how to start to get to a 3 term answer}...
I have this integral
$$\int_{}^{}\frac{1}{{2}^{lnx}} \,dx$$
I'm not sure the best way to do it.
I tried u-substitution:
$u = {2}^{lnx}$ and thus $u = {x}^{ln2}$, therefore $du = ln2({n}^{ln2 - 1}) dx$. However, not sure how to proceed from there.
If I have $\ln\left({a}\right) - \ln\left({b}\right)$ that would equal $\ln\left({\frac{a}{b}}\right)$ or $-(\ln\left({b}\right) - \ln\left({a}\right))$ which is also $- \ln\left({\frac{b}{a}}\right)$. So does this mean $\ln\left({\frac{a}{b}}\right)$ equals $- \ln\left({\frac{b}{a}}\right)$?
I have this sequence:
$${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 -...
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...
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 } }{...
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...
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...
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 --...
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...
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.
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...
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...
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 (...
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...
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!
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...
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(...
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...
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...
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...
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...
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
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...
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...
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...
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...
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
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.
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
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...
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
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 /...
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}
=...
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...
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...