The natural logarithm of x is equal to. Natural logarithm, function ln x

Graph of a function natural logarithm. The function slowly approaches positive infinity as you increase x and quickly approaches negative infinity when x tends to 0 (“slow” and “fast” compared to any power function of x).

Natural logarithm is the logarithm to the base , Where e (\displaystyle e)- an irrational constant equal to approximately 2.72. It is denoted as ln ⁡ x (\displaystyle \ln x), log e ⁡ x (\displaystyle \log _(e)x) or sometimes just log ⁡ x (\displaystyle \log x), if the base e (\displaystyle e) implied . In other words, the natural logarithm of a number x- this is an exponent to which a number must be raised e to get x. This definition can be extended to complex numbers.

ln ⁡ e = 1 (\displaystyle \ln e=1), because e 1 = e (\displaystyle e^(1)=e); ln ⁡ 1 = 0 (\displaystyle \ln 1=0), because e 0 = 1 (\displaystyle e^(0)=1).

The natural logarithm can also be defined geometrically for any positive real number a as the area under the curve y = 1 x (\displaystyle y=(\frac (1)(x))) in between [ 1 ; a ] (\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.)

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.
There is no need to delve into the proof makes a lot of sense, 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, 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 $x=-1$ 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 $)