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Let's solve the given equation:

\[
(3^x)(8^{2x+1}) = 4^{2x+3}.
\]

### Step 1: Express everything in terms of prime bases
We start by rewriting the numbers 8 and 4 as powers of 2:
- \( 8 = 2^3 \),
- \( 4 = 2^2 \).

Substitute these into the equation:

\[
3^x \cdot (2^3)^{2x+1} = (2^2)^{2x+3}.
\]

### Step 2: Simplify exponents
Now, simplify the exponents using the property \((a^m)^n = a^{m \cdot n}\):
- \( (2^3)^{2x+1} = 2^{3(2x+1)} = 2^{6x+3} \),
- \( (2^2)^{2x+3} = 2^{2(2x+3)} = 2^{4x+6} \).

This gives us the new equation:

\[
3^x \cdot 2^{6x+3} = 2^{4x+6}.
\]

### Step 3: Isolate powers of 2
We now equate the powers of 2 and separate the \(3^x\) term. To do this, divide both sides by \(2^{4x+6}\):

\[
3^x = 2^{(4x+6) - (6x+3)} = 2^{-2x + 3}.
\]

Now we have:

\[
3^x = 2^{3 - 2x}.
\]

### Step 4: Take logarithms
To solve this equation, take the logarithm of both sides (logarithms to any base are fine; we will use the natural logarithm):

\[
\ln(3^x) = \ln(2^{3 - 2x}).
\]

Apply the logarithmic power rule, \(\ln(a^b) = b \ln(a)\), to get:

\[
x \ln(3) = (3 - 2x) \ln(2).
\]

### Step 5: Solve for \(x\)
Now solve for \(x\). Distribute \(\ln(2)\) on the right-hand side:

\[
x \ln(3) = 3 \ln(2) - 2x \ln(2).
\]

Move all terms involving \(x\) to one side:

\[
x \ln(3) + 2x \ln(2) = 3 \ln(2).
\]

Factor out \(x\):

\[
x(\ln(3) + 2 \ln(2)) = 3 \ln(2).
\]

Solve for \(x\):

\[
x = \frac{3 \ln(2)}{\ln(3) + 2 \ln(2)}.
\]

### Step 6: Express in the desired form
The problem asks for the answer in the form \(\frac{\log a}{\log b}\), where \(a, b \in \mathbb{Z}\). Notice that \(\ln(2)\) and \(\ln(3)\) are logarithms of integers. The given expression can be rewritten as:

\[
x = \frac{\log 2^3}{\log 3 + \log 2^2}.
\]

Thus, we have:

\[
x = \frac{\log 8}{\log 12}.
\]

### Final Answer:
\[
x = \frac{\log 8}{\log 12}.
\]

Would you like further details or have any questions about this solution?

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Here are 5 related questions for further practice:

1. Solve \(5^x \cdot 25^{2x+1} = 125^{3x+2}\) in a similar form.
2. If \(2^x \cdot 9^{2x} = 27^{x+3}\), solve for \(x\).
3. Simplify the equation \( (6^x)(36^{2x+1}) = 216^{3x+2} \).
4. Solve \( (4^x)(16^{2x+1}) = 64^{3x+2} \) and express \(x\) in the form \(\frac{\log a}{\log b}\).
5. If \(7^x \cdot 49^{x+1} = 343^{2x+3}\), what is \(x\)?

### Tip:
When dealing with exponents, always try to express the terms in prime factored form to simplify the equation.

Find the exact value of x satisfying the equation (3^x)(8^{2x+1}) = 4^{2x+3}. Give your answer in the form loga / logb where a, b ∈ Z.

Learn how to solve the exponential equation (3^x)(8^{2x+1}) = 4^{2x+3} with a step-by-step solution. This involves using logarithms and exponential rules. Suitable for students in Grades 10-12.

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