often take a number e = 2,718281828 . Logarithms by this basis are called natural. When performing calculations with natural logarithms, it is common to operate with the sign ln, not log; while the number 2,718281828 , defining the basis, are not indicated.
In other words, the formulation will look like: natural logarithm numbers X- this is an exponent to which a number must be raised e to get x.
So, ln(7,389...)= 2, since e 2 =7,389... . Natural logarithm of the number itself e= 1 because e 1 =e, and the natural logarithm of unity is zero, since e 0 = 1.
The number itself e defines the limit of a monotone bounded sequence
calculated that e = 2,7182818284... .
Quite often, in order to fix a number in memory, the digits of the required number are associated with some outstanding date. Speed of memorizing the first nine digits of a number e after the decimal point will increase if you notice that 1828 is the year of birth of Leo Tolstoy!
Today there are enough full tables natural logarithms.
Natural logarithm graph(functions y=ln x) is a consequence of the exponential graph as a mirror image of the straight line y = x and has the form:
The natural logarithm can be found for every positive real number a as the area under the curve y = 1/x from 1 to a.
The elementary nature of this formulation, which is consistent with many other formulas in which the natural logarithm is involved, was the reason for the formation of the name “natural”.
If you analyze natural logarithm, as a real function of a real variable, then it acts inverse function to an exponential function, which reduces to the identities:
e ln(a) =a (a>0)
ln(e a) =a
By analogy with all logarithms, the natural logarithm converts multiplication into addition, division into subtraction:
ln(xy) = ln(x) + ln(y)
ln(x/y)= lnx - lny
The logarithm can be found for every positive base that is not equal to one, not just for e, but logarithms for other bases differ from the natural logarithm only by a constant factor, and are usually defined in terms of the natural logarithm.
Having analyzed natural logarithm graph, we find that it exists for positive values of the variable x. It increases monotonically in its domain of definition.
At x → 0 the limit of the natural logarithm is minus infinity ( -∞ ).At x → +∞ the limit of the natural logarithm is plus infinity ( + ∞ ). At large x The logarithm increases quite slowly. Any power function xa with a positive exponent a increases faster than the logarithm. Natural logarithm is a monotonically increasing function, so it has no extrema.
Usage natural logarithms very rational when passing higher mathematics. Thus, using the logarithm is convenient for finding the answer to equations in which unknowns appear as exponents. The use of natural logarithms in calculations makes it possible to greatly simplify a large number of mathematical formulas. Logarithms to the base e are present in solving a significant number of physical problems and are naturally included in the mathematical description of individual chemical, biological and other processes. Thus, logarithms are used to calculate the decay constant for a known half-life, or to calculate the decay time in solving problems of radioactivity. They perform in leading role in many branches of mathematics and practical sciences, they are used in the field of finance to solve a large number of problems, including the calculation of compound interest.
Lesson and presentation on the topics: "Natural logarithms. The base of the natural logarithm. The logarithm of a natural number"
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What is natural logarithm
Guys, in the last lesson we learned a new, special number - e. Today we will continue to work with this number.We have studied logarithms and we know that the base of a logarithm can be many numbers that are greater than 0. Today we will also look at a logarithm whose base is the number e. Such a logarithm is usually called the natural logarithm. It has its own notation: $\ln(n)$ is the natural logarithm. This entry is equivalent to the entry: $\log_e(n)=\ln(n)$.
Exponential and logarithmic functions are inverses, then the natural logarithm is the inverse of the function: $y=e^x$.
Inverse functions are symmetric with respect to the straight line $y=x$.
Let's plot the natural logarithm by plotting the exponential function with respect to the straight line $y=x$.
It is worth noting that the angle of inclination of the tangent to the graph of the function $y=e^x$ at point (0;1) is 45°. Then the angle of inclination of the tangent to the graph of the natural logarithm at point (1;0) will also be equal to 45°. Both of these tangents will be parallel to the line $y=x$. Let's diagram the tangents: 
Properties of the function $y=\ln(x)$
1. $D(f)=(0;+∞)$.2. Is neither even nor odd.
3. Increases throughout the entire domain of definition.
4. Not limited from above, not limited from below.
5. Greatest value No, lowest value No.
6. Continuous.
7. $E(f)=(-∞; +∞)$.
8. Convex upward.
9. Differentiable everywhere.
In the course of higher mathematics it is proven that the derivative of an inverse function is the inverse of the derivative of a given function.
It doesn’t make much sense to go deeper into the proof, let’s just write the formula: $y"=(\ln(x))"=\frac(1)(x)$.
Example.
Calculate the value of the derivative of the function: $y=\ln(2x-7)$ at the point $x=4$.
Solution.
IN general view our function is represented by the function $y=f(kx+m)$, we can calculate the derivatives of such functions.
$y"=(\ln((2x-7)))"=\frac(2)((2x-7))$.
Let's calculate the value of the derivative at the required point: $y"(4)=\frac(2)((2*4-7))=2$.
Answer: 2.
Example.
Draw a tangent to the graph of the function $y=ln(x)$ at the point $х=е$.
Solution.
We remember well the equation of the tangent to the graph of a function at the point $x=a$.
$y=f(a)+f"(a)(x-a)$.
We sequentially calculate the required values.
$a=e$.
$f(a)=f(e)=\ln(e)=1$.
$f"(a)=\frac(1)(a)=\frac(1)(e)$.
$y=1+\frac(1)(e)(x-e)=1+\frac(x)(e)-\frac(e)(e)=\frac(x)(e)$.
The tangent equation at the point $x=e$ is the function $y=\frac(x)(e)$.
Let's plot the natural logarithm and the tangent line. 
Example.
Examine the function for monotonicity and extrema: $y=x^6-6*ln(x)$.
Solution.
The domain of definition of the function $D(y)=(0;+∞)$.
Let's find the derivative of the given function:
$y"=6*x^5-\frac(6)(x)$.
The derivative exists for all x from the domain of definition, then there are no critical points. Let's find stationary points:
$6*x^5-\frac(6)(x)=0$.
$\frac(6*x^6-6)(x)=0$.
$6*x^6-6=0$.
$x^6-1=0$.
$x^6=1$.
$x=±1$.
The point $х=-1$ does not belong to the domain of definition. Then we have one stationary point $x=1$. Let's find the intervals of increasing and decreasing: 
Point $x=1$ is the minimum point, then $y_min=1-6*\ln(1)=1$.
Answer: The function decreases on the segment (0;1), the function increases on the ray $ (\displaystyle ). The simplicity of this definition, which is consistent with many other formulas that use this logarithm, explains the origin of the name "natural".
If we consider the natural logarithm as a real function of a real variable, then it is the inverse function of the exponential function, which leads to the identities:
e ln a = a (a > 0) ; (\displaystyle e^(\ln a)=a\quad (a>0);) ln e a = a (a > 0) . (\displaystyle \ln e^(a)=a\quad (a>0).)Like all logarithms, the natural logarithm maps multiplication to addition:
ln x y = ln x + ln y . (\displaystyle \ln xy=\ln x+\ln y.)
