Problem
WRITTEN RESPONSE Question \# 4: Show all relevant work in the space provided. Consider the equation $16^{3 x}=64^{x-2}$. Olivia and Jackson both solved the equation using different methods. Their work is shown below. Olivia's Solution Line 1: $\left(4^{2}\right)^{3 x}=\left(4^{3}\right)^{x-2}$ Line 2: $4^{5 x}=4^{3 x-6}$ Line 3: $5 x=3 x-6$ Line 4: $2 x=-6$ Line 5: $x=-3$ Jackson's Solution Line 1: $\log 16^{3 x}=\log 64^{x-2}$ Line 2: $3 x \log 16=x-2 \log 64$ Line 3: $2 \log 64=x-3 x \log 16$ Line 4: $2 \log 64=x(1-3 \log 16)$ Line 5: $x=\frac{2 \log 64}{1-3 \log 16}$ a) Olivia made one error in her work, which led to an incorrect answer. State the line in which her error falls and explain her error. (1 mark) b) Algebraically solve the equation again using Olivia's method in the correct way. (2 marks)
Solution
Step 1 ：Olivia's error is in Line 2. She incorrectly simplified \(4^{3x}\) to \(4^{5x}\) and \(4^{3(x-2)}\) to \(4^{3x-6}\). The correct simplification should be \(4^{6x}\) and \(4^{3x-6}\) respectively.
Step 2 ：Let's correct Olivia's method and solve the equation again. We start with the equation \(4^{6x} = 4^{3x - 6}\).
Step 3 ：Since the bases are equal, we can equate the exponents. This gives us the equation \(6x = 3x - 6\).
Step 4 ：Solving this equation gives us \(3x = -6\), which simplifies to \(x = -2\).
Step 5 ：The solution to the equation is \(x = -2\). The other solutions are complex numbers which are not considered in this context.
Step 6 ：Final Answer: \(\boxed{x = -2}\)
