Isomorphism between (R, +) and (R*, ×)

To show that the additive group of real numbers (R, +) is isomorphic to the multiplicative group of non-zero real numbers (R*, ×), we need to find a bijective function that preserves the group structure.

Group Structure of (R, +)
The set of real numbers, denoted by R, forms an additive group under the operation of addition, denoted by +. The group structure of (R, +) is as follows:
1. Closure: For any two real numbers a and b, their sum a + b is also a real number.
2. Associativity: For any three real numbers a, b, and c, (a + b) + c = a + (b + c).
3. Identity Element: The real number 0 acts as the identity element, such that for any real number a, a + 0 = 0 + a = a.
4. Inverse Element: For any real number a, there exists a unique real number -a such that a + (-a) = (-a) + a = 0.
5. Commutativity: For any two real numbers a and b, a + b = b + a.

Group Structure of (R*, ×)
The set of non-zero real numbers, denoted by R*, forms a multiplicative group under the operation of multiplication, denoted by ×. The group structure of (R*, ×) is as follows:
1. Closure: For any two non-zero real numbers a and b, their product a × b is also a non-zero real number.
2. Associativity: For any three non-zero real numbers a, b, and c, (a × b) × c = a × (b × c).
3. Identity Element: The real number 1 acts as the identity element, such that for any non-zero real number a, a × 1 = 1 × a = a.
4. Inverse Element: For any non-zero real number a, there exists a unique real number 1/a such that a × (1/a) = (1/a) × a = 1.
5. Commutativity: For any two non-zero real numbers a and b, a × b = b × a.

Isomorphism
To establish an isomorphism between (R, +) and (R*, ×), we need to find a bijective function f: R → R* that preserves the group structure. In other words, f should satisfy the following conditions:
1. f(a + b) = f(a) × f(b) for all real numbers a and b.
2. f(0) = 1.
3. f(-a) = 1/f(a) for all real numbers a.

One such function that satisfies these conditions is the exponential function, f(x) = e^x, where e is the base of the natural logarithm. The exponential function has the following properties:
1. f(a + b) = e^(a + b) = e^a × e^b = f(a) × f(b) for all real numbers a and b.
2. f(0) = e^0 = 1.
3. f(-a) = e^(-a)