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To prove that the function \( f \), defined as \( f(x) = \log_{10}(x) \), is an isomorphism between the group \( (\mathbb{R}^+, \cdot) \) (the group of positive real numbers under multiplication) and \( (\mathbb{R}, +) \) (the group of real numbers under addition), we need to show two key properties:

1. **Homomorphism**: The function must preserve the group operation, i.e., \( f(ab) = f(a) + f(b) \) for all \( a, b \in \mathbb{R}^+ \).
2. **Bijectivity**: The function must be both injective (one-to-one) and surjective (onto), ensuring that there is a one-to-one correspondence between elements of \( \mathbb{R}^+ \) and \( \mathbb{R} \).

### Step 1: Prove Homomorphism

To prove that \( f \) preserves the group operation, we need to check that for any \( a, b \in \mathbb{R}^+ \),
\[
f(ab) = f(a) + f(b).
\]

Using the definition of \( f(x) = \log_{10}(x) \), we have:
\[
f(ab) = \log_{10}(ab).
\]

By the properties of logarithms, we know:
\[
\log_{10}(ab) = \log_{10}(a) + \log_{10}(b).
\]

Thus,
\[
f(ab) = f(a) + f(b),
\]
which shows that \( f \) is a homomorphism.

### Step 2: Prove Bijectivity

- **Injectivity (One-to-One)**:  
  To show injectivity, assume \( f(a) = f(b) \) for some \( a, b \in \mathbb{R}^+ \). This implies:
  \[
  \log_{10}(a) = \log_{10}(b).
  \]
  By the fact that the logarithmic function is injective (logarithms are one-to-one functions), it follows that:
  \[
  a = b.
  \]
  Therefore, \( f \) is injective.

- **Surjectivity (Onto)**:  
  To show that \( f \) is surjective, we need to prove that for every \( y \in \mathbb{R} \), there exists an \( x \in \mathbb{R}^+ \) such that \( f(x) = y \).  
  Since \( f(x) = \log_{10}(x) \), we need to find an \( x \) such that:
  \[
  \log_{10}(x) = y.
  \]
  By the definition of logarithms, this equation can be solved by setting:
  \[
  x = 10^y.
  \]
  Since \( 10^y \in \mathbb{R}^+ \) for any \( y \in \mathbb{R} \), this shows that \( f \) is surjective.

### Conclusion

Since \( f \) is both a homomorphism and a bijection (i.e., it is injective and surjective), it follows that \( f \) is an isomorphism. Therefore, the group \( (\mathbb{R}^+, \cdot) \) is isomorphic to the group \( (\mathbb{R}, +) \), and the logarithmic function \( f(x) = \log_{10}(x) \) translates problems involving multiplication into problems involving addition.

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Let me know if you need further details! Here are some related questions for deeper understanding:

1. Can you generalize the proof for logarithms with any base \( b \)?
2. What is the importance of the logarithmic function being bijective in group theory?
3. How would this proof change if we worked with logarithms on the complex plane?
4. Can you think of other functions that act as isomorphisms between different algebraic structures?
5. How does the concept of a homomorphism apply to other mathematical structures, such as rings or fields?

**Tip**: In general, proving bijectivity involves checking injectivity and surjectivity separately. For injectivity, assume two elements map to the same output; for surjectivity, show that for any output, there is a corresponding input.

Learn how to prove that the function f(x) = log_10(x) is an isomorphism between the group of positive real numbers under multiplication and the group of real numbers under addition. This step-by-step solution covers key group theory concepts, including homomorphisms, injectivity, and surjectivity.

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