Sure, let's solve the equation step-by-step to find the value of [tex]\(a\)[/tex].
Given equation:
[tex]\[2^{(x-4)} = 4 \cdot a^{(x-6)}\][/tex]
We can rewrite the number 4 as a power of 2:
[tex]\[4 = 2^2\][/tex]
Substitute this back into the equation:
[tex]\[2^{(x-4)} = 2^2 \cdot a^{(x-6)}\][/tex]
This gives us:
[tex]\[2^{(x-4)} = 2^2 \cdot a^{(x-6)}\][/tex]
Now, let's express the equation in terms of powers of 2:
[tex]\[2^{(x-4)} = 2^{(2)} \cdot a^{(x-6)}\][/tex]
We can equate the exponents of 2 on both sides, but we need to isolate the term involving [tex]\(a\)[/tex]. To do this, let's consider [tex]\(a\)[/tex] in terms of base 2.
Express the given equation as:
[tex]\[2^{(x-4)} = 2^{(2 + (x-6)\log_2(a))}\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[x - 4 = 2 + (x - 6)\log_2(a)\][/tex]
First, simplify the exponent equation:
[tex]\[ x - 4 = 2 + (x - 6)\log_2(a) \][/tex]
Subtract [tex]\(x - 6\cdot\log_2(a)\)[/tex] from both sides:
[tex]\[ x - 4 - 2 = (x - 6)\cdot\log_2(a) \][/tex]
Combining like terms:
[tex]\[ x - 6 = (x - 6)\cdot\log_2(a) \][/tex]
Since [tex]\(x - 6\)[/tex] appears on both sides, we can factor it out:
[tex]\[ x - 6 = (x - 6)\cdot\log_2(a) \][/tex]
This equation holds if:
[tex]\[ \log_2(a) = 1 \][/tex]
Solving for [tex]\(a\)[/tex]:
[tex]\[ \log_2(a) = 1 \][/tex]
[tex]\[ a = 2 \][/tex]
Thus, the value of [tex]\(a\)[/tex] is:
[tex]\[a = 2\][/tex]
So, the correct answer is [tex]\( \boxed{2} \)[/tex].
