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.
when I differentiate:
$$ln4x+sin(x)$$
I get:
$$\frac{1}{x}+cos(x)$$
and Wolfgram agrees
But then when i test this by calculating indefinite integral, I get:
$$ln(x)+cos(x)$$
Which leaves me with three questions:
1. what happened to the 4?
2. why isn't it integrating back to (at least)...
Homework Statement
Find the derivative of y = x^2 + x^(2x)The Attempt at a Solution
By looking at the equation I think I need to use implicit differentiation + natural logs. But I can't do anything with:
lny = ln(x^2 + x^(2x))
So I assume I'm wrong.. Any help??
Homework Statement
What does it mean when the derivative of a function f(x) is in the form:
d ln f(x) / d ln x
?
Is it the logarithmic scale derivative, or something?
Homework Equations
d ln f(x) / d ln x
The Attempt at a Solution
Googling.
Hello,
The problem is ∫(ln x)/(x + x ln x) dx.
I've done most other problems in the set, but don't know where to start with this one. Although we are just learning integration by parts, I'm not sure how this would apply. I can get to ∫u/(1+u) du
Thanks for any help.
Homework Statement...
Homework Statement
I have a problem that I'm working on that I have almost solved, yet I am just a tad off of what the book says the answer is. I will show the way I'm doing it, and where I depart from the steps the book takes.
The graph of y=(x^2)^x has two horizontal tangent lines. Find...
So I've come across a derivative problem that I need to solve that is showing me some of my weaknesses in my understanding/solving of Ln and e. This is what I've done so far.
\(Q = 350\frac{1}{2}^{(\frac{t}{13.1})}\)
\(Q = 350 * \frac{1}{2}^{(\frac{t}{13.1})}\)
\(ln{Q} = \ln{350} *...
Homework Statement
Find y' of
y= 1-3ln(7x)/x^4
Homework Equations
The Attempt at a Solution
I used the quotient rule and got:
y'=x^4*d/dx(1-3ln(7x)-(1-3ln(7x)*d/dx(x^4)/(x^4)2
which is: x^4*(0-3*1/7x*7)-(1-3ln(7x))*4x^3/x^8
simplified to: 3x^4/x-1+3ln(7x)*4x^3
3x^3-4x^3+12x^3ln(7x)/x^8
take...
Determine the infinite limit.
lim x->4+ ln(x^2-16)
I know from graphing the equation and doing a table that the limit is -infinity, but my book is saying to do the following.
Let t = x^2-16, Then as x->4+, t->0+, and lim x->4+ ln(x^2-16)=lim t->0+ ln(t) by 3
Homework Statement
Solve the Equation, Check for extraneous solutions.
(ln x)^2 = ln (x^2)
Homework Equations
-None-
The Attempt at a Solution
I tried simplifying it down for myself but got stuck and don't know what to do next.
(ln x)^2= ln x^2
(loge x)^2 = loge x^2...
Homework Statement
Use the form of the definition of the integral to evaluate the following:
lim (n \rightarrow ∞) \sum^{n}_{i=1} x_{i}\cdotln(x_{i}^{2} + 1)Δx on the interval [2, 6]
Homework Equations
x_{i} = 2 + \frac{4}{n}i
Δx = \frac{4}{n}
Ʃ^{n}_{i=1}i^{2} =...
I'm relatively new to C programming and I am trying to make a program to take and print the natural log of a number input by the user but for some reason my program always returns incorrect answers. For example I entered 1 as my number, and the answer came back as some ridiculously big value...
Hi.
I just saw on wikipedia that natural logarithm has such a property:
[x/(1+x)] < ln (1 + x) < x
(http://en.wikipedia.org/wiki/Natural_logarithm)
Can anyone pls tell me how to prove this?
Proving [x/(1+x)] and ln (1 + x) less than 'x' is easy.. But how abt [x/(1+x)] < ln (1 + x)...
[PLAIN]http://img31.imageshack.us/img31/9004/screenshot20111117at720.png
Proofs always get to me for some reason. It's like other problems I can do, but when it comes to proofs I don't know what to put. Can anyone show me steps?
Thank you
It seems this is done randomly, yet it must not be random, there must be some logic to it, but when do you know to do it? What is like a clue or the rules that indicate you can/should take the natural log of both sides of an equation?
Any information on the following formula for natural logarithm (I looked in wikipedia and Mathworld but didn't see it). It came from another equation I was working on a bit ago, and I was curious about it as I didn't recall seeing it before (which doesn't mean I haven't), although it reminded...
Is the following relationship true?:
\frac{\partial (ln(k))}{\partial P}=\frac{1}{k}\frac{\partial k}{\partial P}
I am getting both of these terms from a paper on mineral physics and they seem to use both terms interchangeably. If so, how are these related?
Homework Statement
Calculate the following:
\int^{-4}_{-6} (x^-1+5x)dx
Homework Equations
The Attempt at a Solution
I've worked this down to
ln(-4) + 5(-4) - ln(-6) - 5(-6)
=ln(-4)-ln(-6)+10
The answer to the left half (the ln parts) of the equation is undefined, and the...
Homework Statement
Find the integral to the following expression
\int\frac{1}{x-1}dx
Homework Equations
\int udv = uv - \int vdu
\int\frac{1}{x}dx = ln(x)
The Attempt at a Solution
Given the information above, would a correct answer to this problem be ln(x-1) ? If not...
Homework Statement
Differentiate, f(x)=71+x/1-x
Homework Equations
The Attempt at a Solution
Working this out, hoping it's right as I was going along, my final answer is 71+x/1-x * (2ln7)/1-x
natural log helpp!
Homework Statement
ln(x+10)+ln(x+4)=2
2. The attempt at a solution
okay so i took it down to (-14+-(14^2 - 4(e^2 - 40))^1/2) / 2
which i broke down to 356 - 4e^2 inside the sqrt.
am i on the right track and if so Whats the next step?
so I've got this equation down to
x*ln(2/x) = 5
Can someone show/point me to a page where I can see how to solve this. I am under the impression it requires some type of ratio computation.
thanks!
SORRY FOR THE LACK OF FORMATTING, I SPENT 30 MINUTES TRYING TO FORMAT AND IT KEPT GETTING MESSED UP.
Homework Statement
Evaluate by substitution.
Integral of dx/xln(x^2)
Homework Equations
integral of 1/u du = ln(u) + C
The Attempt at a Solution
u = ln(x^2)
du = 2/x...
Homework Statement
Evaluate the derivative of the following function:
p(x) = (5x)-ln(5x)
Homework Equations
\frac{d}{dx}lnx = \frac{1}{x}
Not sure what else...
The Attempt at a Solution
I know that I will have to use the chain rule in this problem, but actually...
Homework Statement
Find the area under the given equation using definite integration.
y = 4ln(3-x) from 0 to 2
The Attempt at a Solution
Integration from 0 to 2
4 (integration from 0 to 2) (3-x)
Let u = 3-x
du = -dx
-du = dx
-4(integration from 0 to 2) du
Idk where to go from there :/...
Well I've been doing fairly complex thermodynamics in classical mechanics lectures with a bit of medium to advanced class, first of all , it took me a very long time to wrap my head around the concept of Entropy, however I still don't understand why we use ln (Natural Log) to calculate the...
Homework Statement
what is the integral of ln(e2x-1) dx
2. The attempt at a solution
I got
u= ln(e2x-1)
du = 2e2x-1 \ e2x-1
du = 2 dx
1/2(integral of u du)
1/2 ln (e2x-1 ) +c
=
1/4 [ln (e2x-1 )]2 + c
Could you please let me know if this is correct? I don't know how to integrate ln...
Homework Statement
Given the Equation ln(x+1)-ln(x)=0 find x (If it exists.
Homework Equations
The Attempt at a Solution
ln(x+1)-ln(x)=0
ln(x+1)=ln(x)
x+1 = x
1 = x-x
Solution does not exist. (My book "agrees" with me in the solutions.)
However, typing up the...
Homework Statement
(1)
Real variables x and y are related by the equation
3ln(y +4) = 2ln(x +2) − 2ln(x +9)+3ln(x^2 − 1).
Determine the range of values of x and y for which the expressions on each side of this equation are defined.
Find y explicitly as a function of x, that is...
Prove that sum 1/2^(n+1)*n/(n+1), from n=0 to infinity, converges to 1 - log(2), where log stands for the natural logarithm.
I know that the Taylor series for log(x) about x=1 is sum (-1)^(n+1)*(x-1)^n/n, but I don't see how these two statements are consistent.
Thanks for any pointers!
Homework Statement
My homework problem is the double integral of y/1+xy dxdy. It is a definite double integral and both integrands have the values of a = 0 and b = 1. Homework Equations
Integration by parts: uv - int(vdu)
The Attempt at a Solution
My first step of the double integral is I...
Homework Statement
Evaluate the integral when x > 0:
indefinite integral of ln(x2+19x+84)dxHomework Equations
I know I need to use some form of integration by parts: integral of u*dv=uv-(integral of(du*v))
The Attempt at a Solution
I began by making u=ln(x2+19x+84) and dv=dx. Thus, (after...
Homework Statement
ln(3x+1) = 3-lnx Homework Equations
Solve for xThe Attempt at a Solution
Well I put the ln on the left side
ln(3x+1)+ln(x) = 3
Then I combine them
ln ((3x+1)(x)) = 3
So I take e
e^ln(x(3x+1)) = e^3
I get x(3x+1) = e^3
So I divide the x
(3x+1)/(x) = (e^3)/(x)...
the question is simple but i can't seem to think of a solution.
for what b>0 does ln(x)=b x^{2} have exactly one solution, and not 0.
I've tried playing with ln rules but can't seem to think of a solution
Hello,
Consider x \in (0,1) , that is x between 0 and 1. Can someone explain why the following is true:
\frac{x-1}{x+1} = \tanh \left( \ln \left( \frac{x}{2} \right) \right)
Homework Statement
using the following definition: ln x = \int dx/x, give best possible upper and lower estimates of ln n, n a positive integer.
then, use this result to show that the limit of the function as n approaches infinity is infinity.
Homework Equations
--
The Attempt...
Hello, I am fairly new here, so thank you in advance for your help. This is not a homework problem, just one of curiosity based on my limited knowledge of asymptotic expansions. I'm curious about how the function ln(x) or ln(1+x) behaves for large values of x. Plotting in Matlab, I see a linear...
Homework Statement
Trying to solve for K:
ln(1-4k)=-6k
Homework Equations
The Attempt at a Solution
I know that need to take the e of both side, to get 1-4k=e^-6k, but I cannot remember any properties of e to allow me to remove the k from the exponent. Any tips is appreciated.
Homework Statement
\int^{4}_{2} \frac{dx}{x\left(lnx\right)^{2}}
Homework Equations
Let u=lnx
du=\frac{1}{x}dx
x=2 \rightarrow u=ln2
x=4 \rightarrow u=ln4
The Attempt at a Solution
so with the u substitution we have:
\int^{ln4}_{ln2} \frac{1}{u^{2}}du
which goes to...
Homework Statement
Solve ln(sinh-1(x)) = 1
(Just give the formula for x, no calculator is necessary)
sinh-1(x) = asinh
Homework Equations
The Attempt at a Solution
When I did this, I knew sinh-1(x) = ln(x + sqrt.(x^2 +1), but the equation would look like
ln(ln(x +...
Homework Statement
This is my first time ever posting anything on here...but we just started working with ln? I know that it's the base e? or something like that...
but the problem is this...
1/3=ln(x^2/x-4)Homework Equations
The Attempt at a Solution
I haven't attempted..i...
1. Graphing Carefully: Sketch f(x)= 10(ln(ln(x))/ln(x) accurately using calculus, your calculator, or both to aid you. Make sure to include all vertical asymptotes, as well as local extrema
2. f(x)= 10(ln(ln(x))/ln(x)
3. OK. I might come across as stupid but here goes... I don't...
Homework Statement
y and x are both close to 1.
ln(y) - ln(x) \approx (y-x)(lnx)'
Can someone explain me why this is true?
By the way, (lnx)' is the derivative of lnx, which is just 1/x
The Attempt at a Solution
I guess this is something to do with the taylor series of ln(x). I tried to...
Preparing for the math test, and cannot understand what to do for one of my practice problems:
I need to find if the series is convergent or divergent, using test for divergence (divergent if limit doesn't equal 0)
sigma from 1 to infinity of ln((n^2+1)/(2n^2+1))
I see that I should use...