This problem involves solving the exponential equation:

$2^{2x+4} = 3^{x-3}$

Step-by-Step Solution:

1. Take the natural logarithm (ln) on both sides: $\ln(2^{2x+4}) = \ln(3^{x-3})$

2. Apply the logarithmic power rule ($\ln(a^b) = b \ln(a)$): $(2x + 4) \ln(2) = (x - 3) \ln(3)$

3. Expand both sides: $2x \ln(2) + 4 \ln(2) = x \ln(3) - 3 \ln(3)$

4. Rearrange to isolate $x$: Group all $x$-terms on one side: $2x \ln(2) - x \ln(3) = -4 \ln(2) - 3 \ln(3)$

   Factor $x$ on the left-hand side: $x \left( 2 \ln(2) - \ln(3) \right) = -4 \ln(2) - 3 \ln(3)$

5. Solve for $x$: $x = \frac{-4 \ln(2) - 3 \ln(3)}{2 \ln(2) - \ln(3)}$

   This is the solution expressed in terms of natural logarithms.

Decimal Approximation:

Using a calculator:

• $\ln(2) \approx 0.6931$
• $\ln(3) \approx 1.0986$

Substitute: $x = \frac{-4(0.6931) - 3(1.0986)}{2(0.6931) - 1.0986}$

1. Calculate the numerator: $-4(0.6931) - 3(1.0986) = -2.7724 - 3.2958 = -6.0682$

2. Calculate the denominator: $2(0.6931) - 1.0986 = 1.3862 - 1.0986 = 0.2876$

3. Divide: $x = \frac{-6.0682}{0.2876} \approx -21.1$

Final Answer:

• Solution in terms of natural logarithms: $x = \frac{-4 \ln(2) - 3 \ln(3)}{2 \ln(2) - \ln(3)}$
• Decimal approximation: $x \approx -21.1$

Would you like further clarification or additional steps?

Related Questions:

1. What are the rules for simplifying logarithmic expressions?
2. How do you handle equations involving two different bases like in this example?
3. Why are logarithms used to solve exponential equations?
4. What are common properties of logarithms used in such equations?
5. How can this result be verified graphically?

Tip:

Always check the base of the exponential terms when solving such equations; if they differ, logarithms are often essential!