Why do degrees behave this way?

Question

Mathematics

Let's say we have addition a+b=c. We know a,b and find c. We can know a and c and find b: b=ca. And we can know b and c and find a: a=cb. That is, there is an action "addition" and two other derivatives, subtractions, but they are the same

Let's say we have a multiplication ab=c. We know a,b and find c. We can know a and c and find b: b=c/a. And we can know b and c and find a: a=c/b. That is, there is the action "multiplication" and two other, derivatives, divisions, but they are

the same. But with a degree, it is not so. Let's say a^b=c. We know a,b and find c. If we know b and c, we find a: a= root of c to power b. If we know a and c we find b: b = logarithm to base a of c. Here, too, there is an action and two derivatives. But why are they the same in addition and multiplication, but in powers did you have to invent two actions - the root and the logarithm?

Answer 1

Because the degree is not commutative.
a + b = b +a
a*b = b*a
a^b != b^a

Answer 2

Logarithms have unique properties that have determined their widespread use to significantly simplify time-consuming calculations. In the transition “to the world of logarithms”, multiplication is replaced by a much simpler addition, division by subtraction, and exponentiation and root extraction are converted, respectively, into multiplication and division by an exponent. Laplace said that the invention of logarithms, "by reducing the labor of the astronomer, doubled his life."

Wiki .
To simplify the calculation, there is also the theory of residues ...

Answer 3

>Let's say we have multiplication ab=c. We know a,b and find c. We can know a and c and find b: b=c/a. And we can know b and c and find a: a=c/b. That is, there is a “multiplication” action and two other, derivative, divisions, but they are the same
a * b \u003d c
b \u003d 0, c \u003d 0, what is a? =)