To solve the equation \(8^x \cdot 4^{x+1} = 48\), we need to express all terms with the same base and then solve for \(x\). Here are the steps:

1. Rewrite \(8\) and \(4\) as powers of \(2\):
   \[ 8 = 2^3 \]
   \[ 4 = 2^2 \]

2. Substitute these values into the equation:
   \[ (2^3)^x \cdot (2^2)^{x+1} = 48 \]

3. Simplify the exponents:
   \[ 2^{3x} \cdot 2^{2(x+1)} = 48 \]
   \[ 2^{3x} \cdot 2^{2x+2} = 48 \]

4. Combine the exponents (since the bases are the same, we add the exponents):
   \[ 2^{3x + 2x + 2} = 48 \]
   \[ 2^{5x + 2} = 48 \]

5. Express \(48\) as a power of \(2\):
   \[ 48 = 2^5 \cdot 3^1 \]

Since \(48\) has a factor of 3, it can't be expressed as a single power of 2. Instead, we compare the exponents directly by taking the logarithm base 2 of both sides:

6. Isolate the exponential term:
   \[ 2^{5x + 2} = 48 \]

7. Take the natural logarithm (or log base 2) of both sides:
   \[ \log_2(2^{5x + 2}) = \log_2(48) \]
   \[ 5x + 2 = \log_2(48) \]

8. Calculate \(\log_2(48)\):
   \[ \log_2(48) = \log_2(2^5 \cdot 3) = 5 + \log_2(3) \]

9. Substitute back:
   \[ 5x + 2 = 5 + \log_2(3) \]

10. Solve for \(x\):
    \[ 5x + 2 = 5 + \log_2(3) \]
    \[ 5x = 3 + \log_2(3) \]
    \[ x = \frac{3 + \log_2(3)}{5} \]

So the solution for \(x\) is:
\[ x = \frac{3 + \log_2(3)}{5} \]

Would you like more details or have any questions?

Here are 8 related questions:

1. How do you convert other bases to powers of 2?
2. What is the logarithmic property used in step 7?
3. How can you solve an equation with different bases?
4. What are the properties of exponents that help in solving such equations?
5. How do you convert from natural logarithm to logarithm base 2?
6. What if the bases are not directly convertible to powers of 2?
7. How would you solve if the equation was \(8^x \cdot 4^{x+1} = 50\)?
8. What is the significance of the factor 3 in the solution?

**Tip:** When solving exponential equations, expressing all terms with the same base often simplifies the problem significantly.

Learn how to solve the exponential equation 8^x * 4^(x+1) = 48 step-by-step with detailed explanations. This problem involves converting bases to powers of 2, using logarithmic properties, and understanding the properties of exponents. Suitable for advanced high school students.

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