(ab)^(log a + log b) = a^log a . b^log b . a^2logb

Let's simplify and solve the given equation step by step.

The equation is:

\[
(ab)^{(\log a + \log b)} = a^{\log a} \cdot b^{\log b} \cdot a^{2\log b}
\]

### Step 1: Simplify the left-hand side (LHS)

Using the logarithm property \(\log(ab) = \log a + \log b\), we rewrite the LHS:

\[
(ab)^{(\log a + \log b)} = ab^{\log(ab)}
\]

Since \(\log(ab) = \log a + \log b\), we have:

\[
(ab)^{(\log a + \log b)} = (ab)^{\log(ab)}
\]

### Step 2: Simplify the right-hand side (RHS)

Now, let's simplify the RHS:

\[
a^{\log a} \cdot b^{\log b} \cdot a^{2\log b}
\]

We can combine the terms involving \(a\):

\[
a^{\log a} \cdot a^{2\log b} = a^{\log a + 2\log b}
\]

So the RHS becomes:

\[
a^{\log a + 2\log b} \cdot b^{\log b}
\]

### Step 3: Equating LHS and RHS

Now, we equate the LHS and RHS:

\[
(ab)^{\log(ab)} = a^{\log a + 2\log b} \cdot b^{\log b}
\]

Notice that the equation isn't immediately clear if it's an identity, so let's explore whether the forms can be made identical or if this indicates an equality only under certain conditions.

For \(ab^{\log(ab)}\), expanding using properties of exponents:

\[
(ab)^{\log(ab)} = a^{\log(ab)} \cdot b^{\log(ab)}
\]

Now compare this with the right-hand side:

\[
a^{\log a + 2\log b} \cdot b^{\log b}
\]

The equation holds true when both sides match under the conditions where the exponents result in the same powers for \(a\) and \(b\).

So the equation seems to be an identity under specific conditions, particularly when logarithmic terms balance each other out in the exponentiation.

Would you like more details on specific steps, or is there any part of this you'd like to explore further?

### Related Questions
1. How do logarithmic identities simplify complex expressions?
2. What conditions are necessary for such logarithmic identities to hold?
3. Can we find specific values of \(a\) and \(b\) where this identity simplifies?
4. How does changing the base of the logarithm affect such expressions?
5. How can we apply these identities in solving exponential equations?

### Tip:
Always verify logarithmic identities by exploring their domain of validity and any conditions necessary for equality to hold.

Learn how to simplify and solve the exponential equation (ab)^(log a + log b) = a^log a . b^log b . a^2logb step-by-step. Explore the application of logarithmic identities and exponentiation in solving complex equations. Suitable for advanced mathematical study.

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