The problem asks us to solve the equation $2^{x+2} = 81$.

Solution:

1. Rewrite the equation: $2^{x+2} = 81$

2. Take the logarithm of both sides: Using the natural logarithm ($\ln$), we get: $\ln(2^{x+2}) = \ln(81)$

3. Apply the logarithmic power rule ($\ln(a^b) = b\ln(a)$): $(x+2) \ln(2) = \ln(81)$

4. Solve for $x+2$: Divide both sides by $\ln(2)$: $x+2 = \frac{\ln(81)}{\ln(2)}$

5. Simplify further: Evaluate the logarithms: $\ln(81) = \ln(3^4) = 4 \ln(3)$ So: $x+2 = \frac{4\ln(3)}{\ln(2)}$

6. Isolate $x$: Subtract 2 from both sides: $x = \frac{4\ln(3)}{\ln(2)} - 2$

7. Final calculation (optional numeric value): Using approximations: $\ln(3) \approx 1.0986, \quad \ln(2) \approx 0.6931$ Substitute these values: $x = \frac{4(1.0986)}{0.6931} - 2$ $x \approx \frac{4.3944}{0.6931} - 2$ $x \approx 6.34 - 2$ $x \approx 4.34$

Final Answer:

$x \approx 4.34$

Would you like more details or further clarifications?

Related Questions:

1. How do we solve similar exponential equations when the base isn't given as 2?
2. What if the base is a fraction? How does it change the solution?
3. How do logarithmic rules apply to equations with multiple exponents?
4. What happens if $81$ were expressed as a power of 2 (approximation)?
5. How can this solution be verified using a calculator?

Tip:

When solving exponential equations, always isolate the exponential term before applying logarithms for simplicity.