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:
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{\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:
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{\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.
good day
I want to study the convergence of this serie and want to check my approch
I want to procede by asymptotic comparison
artgln n ≈pi/2
n+n ln^2 n ≈n ln^2 n
and we know that
1/(n ln^2 n ) converge so the initial serie converge
many thanks in advance!
So far, I found the derivative of ##f##:
\begin{align*}
\frac{d}{dx}\,f(x)&=&-\frac{d}{dx}\,\ln(-x)\\
&=&-\left(\frac{1}{(-x)}\right)(-1)\\
&=&-\frac{1}{x}
\end{align*}
##f'(x)## is always positive and never zero on its domain.
Hence, ##f## does not have a local maximum and is always...
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Solve
$$\displaystyle\lim_{h\to 0}
\dfrac{\ln{(4+h)}-\ln{h}}{h}$$
$$(A)\,0\quad
(B)\, \dfrac{1}{4}\quad
(C)\, 1\quad
(D)\, e\quad
(E)\, DNE$$
The Limit diverges so the Limit Does Not Exist (E)ok the only way I saw that it diverges is by plotting
not sure what the rule is that observation...
Hi, I'm currently taking Chemistry 101 and came across this equation that seems to contradict what I've learned before. I don't know the name of it, but here is the equation and its implication.
Now another equation we have learned is the Arrhenius equation, which is as follows:
If I...
Homework Statement
What is the power series for the function ln (x+1)? How do you find the sum of an infinite power series?
Homework Equations
sigma from n=1 to infinity (-1)^n+1 (1/n2^n)
That is the power series, how is that equivalent to ln (x+1)?
How do you find the sum, or what does it...
With S = K ln W where W is probability system is in the state it is in relative to all other possible states :
W = VN , V = volume, N = number of particles so ln W = N (ln V)
And this expression is for non equilibrium state.
For equilibrium state S = K ln Ω Then is the only difference between...
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...
Hi, todos:
Do you know how to calculate the definte integral for Integral for ##\int_{-1}^{1} [P_{l}^{m}]^2 \ln [P_{l}^{m}]^2 dx##, where ##P_{l}^{m} (x)## is associated Legendre functions. Thanks for your time and help.
In the image below, why is the third line not \frac {ln(cosx)} {sinx}+c ? Wouldn't dividing by sinx be necessary to cancel out the extra -sinx that you get when taking the derivative of ln(cosx)? Also, wouldn't the negatives cancel?
I tried with Google but I couldn't find anything, so here goes: When I "use ln on a quantity" (I don't really know how to phrase it in english, as we just have a verb for it), say, I have n = 0.00149 kg/m*s, and I put it into the ln, so now I have ln(0.00149 kg/m*s) what happens to the SI Units...
Homework Statement
for the domain of ln (x^2 + y^2 ) , it it given in my notes that the ans is x ≠ 0 and y ≠ 0
IMO , it's wrong to give x ≠ 0 and y ≠ 0 , because the meaning of x ≠ 0 and y ≠ 0 is that x and y can't be 0 all the times so just leave the ans (x^2 + y^2 ) > 0 , will do ...
Homework Statement
the given ans is x ≠ 0 , y ≠ 0 , the ans given is x ≠ 0 , y ≠ 0 . I don't understand the ans , why the author leave the ans like this ?
IMO , x can be 0 as long as y not = 0
y also can be 0 , as long as x not = 0 ,by giving the ans in x ≠ 0 , y ≠ 0 , the author just rule out...
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 have this equality:
$$ (\ln\left({n}\right))^4 < {n}^{\frac{1}{4}} $$ where $ n > 1$
Can I derive a law from this such that
$$ (\ln\left({n}\right))^b < {n}^{\frac{1}{b}} $$ where $n > 1$ ?
If I have this sequence
$$a_n = \ln\left({\frac{n}{n^2 + 1}}\right)$$
I need to find:
$$ \lim_{{n}\to{\infty}} \ln\left({\frac{n}{n^2 + 1}}\right)$$
Shouldn't I be able to find the limit of$$ \lim_{{n}\to{\infty}} \frac{n}{n^2 + 1}$$ (which is $0$) and then substitute the result of that...
Homework Statement
How is ## e^log√(1-x^2)## equal to ##√(1-x^2)?##Homework EquationsThe Attempt at a Solution
taking ln on the function, ln√(1-x^2). lne⇒ ln√(1-x^2) ....
Hello there!
There is a problem with calculating the uncertainty from semi- ln plot. The linear fitting gives standard errors as you can see in attached picture. In the Y axis are ln J values, obviously. If the intersection with y-axis, x=0, then we get the point y=b=-33,21, and it's ln J', so...
My book finds a function of x say ln(x). It is the area under 1/x. Having the properties (d/dx) ln x = 1/x and ln 1 = 0. It says it determines ln(x) completely. It satisfies the laws of logarithms, but why can I regard it as a logarithm just because it satisfies those laws?
Hi everyone,
Weinberg uses spatial translation invariance to derive the momentum operator. But the way he does it puzzles me. Here is an excerpt of the book.
Equation 3.5.1 is the definition of the unitary operator ##U(x)## for translation invariance:
$$U^{-1}(x)XU(x) = X+x,$$ with...
Since lnx is defined for positive x only shouldn't the derivative of lnx be 1/x, where x is positive. My books does not specify that x must be positive, so is lnx differentiable for all x?
Hey guys, I have a question concerning the rewriting of a differential equation solution.
In the example above, they rewrite [y=(plus/minus)e^c*sqrt(x^2+4)] as [y=C*sqrt(x^2+4)]. I understand that the general solution we get as a result represents all the possible functions, but if we were to...
An old book I have on elementary Statistical Mechanics (Rushbrooke) uses as an especially simple case a system with one energy level. This level is doubly degenerate. The author doesn't give an example of such a system. Can anyone think of one? And would it have entropy k\ ln 2?
[My thoughts...
Does\: $ \sum_{1}^{\infty} ln(1+\frac{1}{n}) $\: converge?
I tried the limit comparison test with bn=1/n and got that it diverges, which also looks right.
However I also tried the ratio test:
$ \lim_{{n}\to{\infty}} \left| \frac{{a}_{n+1}}{{a}_{n}} \right| = \lim_{{n}\to{\infty}} \left|...
Greetings,
I have some questions about ln(x) and e^x graphs , with figuring out Domain , range and line of asymptote.
Q1) How can I know if this graph is ln(x) or e^x
(I thought it was e^x graph since there's no x-axis intercept , however the answer in marking scheme is:
Domain : xεR , x>-3...
This is really strange: If i rewrite a potens function i get a function which should not be possible (it does not give the same values). What did i do wrong?
Y = b*X^a
lnY = ln(b*X^a)
lny = lnb + ln X^a
lnY = lnb + a*lnx
we raise both sides to e
Y = b+ e^(a*lnx)=b+(...
Homework Statement
Dear all
I am calibrating a temperature measurement model and I am stuck with an equation. The variable z is given; x and y represent two regression terms with common regressors - which I will solve for a specific regressor in a second step.
Homework Equations...
I have to say if each reactant is first, second or zeroth order.
Now, I know that usually, we have plots of ln([]) over time. But my teacher wants to trick me.
Here is how I do this:
Take two data points: convert them to [ ] and normal rate (remove the ln() function).
Compare the two...
Homework Statement
y=ln(tan x), dy/dx=2/(sin2x) and d2y/dx2= -4(cos 2x)/(sin 2x)^2 , show that d3y/dx3 = ((4)(3+cos 4x))/(sin 2x)^3 ... i got the solution for dy/dx and d2y/dx2 but not d3y/dx3 , can anyone show me how to get d3y/dx3 please? Thanks in advance!
Homework Equations
The...
Hello all,
I am trying to solve this integral,
\[\int \ln(x^{2}-1) \, dx\]
but I get stuck no matter what I do, if I go for substitution or parts...
thanks
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!
Doing some self prep for Diff EQ starting next week.
Determine the decay rate of C14 which has a 1/2 life of 5230. Using e^kt as a function,
I solve using k5230=ln.5 which gives the obvious answer of negative what I want. How do I know to use the reciprocal (ln2) other than to "just know" I...
Homework Statement
If you have, for example, 2 + 4 - 1, you can get the answer (5), by doing both:
= 2 + (4 - 1)
and,
= (2 + 4) - 1
But the same logic does not work with logs: to get the right answer (4/3) here you must do:
=(ln(9/4) + ln (16/9)) - ln (3/1)
and NOT...
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...