What Is the Natural Logarithm?

The natural logarithm is the logarithm of the constant e as the base and is denoted as lnN (N> 0). It has important significance in natural sciences such as physics and biology, and the general expression method is lnx. It is also common in mathematics to use logx to represent natural logarithms.

The concept of logarithms began in 1614,

The meaning of the constant e is what can be achieved in a unit of time, continuous double growth

Problem: Finding Complex Numbers

Logarithm

for

The principal value of the angle.

answer:

With plural

Through the exponential function

will

Maps to

.

From the definition of plural equality, we get:

and so

, which is

Remember

As a logarithmic function, you can see that in a complex number, a logarithmic function is a multivalued function (that is, one independent variable corresponds to multiple dependent variables), and there are an infinite number of branches. In particular, when k = 0, we say

Is the principal value of the logarithmic function

To represent.

That is, the real part of w is the modulus of z taking the natural logarithm, and the imaginary part is the principal value of the angle of z. This is the logarithmic formula when the true number is complex. Note that because the real part requires a natural logarithm to the modulus of z, r 0. We know that in the complex plane, only 0 has the modulus of 0, and the modulus of any other complex number is greater than 0, so in the complex field, all complex numbers except z = 0 can be logarithmic.

Example: Find ln (-1)

Solution: -1 = cos + isin, its modulus is 1, and the principal value of the amplitude is . Substituting into the formula:

From this

, which is

This is Euler's identity. ^[2]