Final answer:
To solve 4ˣ⁺²=12 for x, we use the change of base formula to convert to base 10 logarithms, apply logarithm properties, and then compute the value of x using a scientific calculator. The answer is b. -0.207519.
Step-by-step explanation:
To solve the equation 4ˣ⁺²=12 for x, we can take the logarithm of both sides. It's useful to apply the change of base formula, which is logᵃy = log y / log b. This allows us to convert the base 4 logarithm to a more familiar base, such as 10 or e (natural logarithm).
Let's proceed with the following steps:
- Take the logarithm of both sides using base 10: log(4ˣ⁺²) = log(12).
- Apply the change of base formula: log(4ˣ⁺²) = log(12) / log(4).
- Express x⁺² as a product of logs: log(4ˣ) + log(4²) = log(12) / log(4).
- Since 4² is 16, and we can simplify this to 2log(4ˣ) = log(12) / log(4) - log(16) / log(4).
- Solve for log(4ˣ) and then for x: log(4ˣ) = (log(12) - log(16)) / 2log(4), so x = (log(12) / log(4) - log(16) / log(4)) / (2log(4)).
- Use a scientific calculator to compute the value of x.
Once computed, we compare the result to the multiple-choice options provided. Based on calculation, the answer is (b) -0.207519.