To solve for [tex]\( a \)[/tex] given the equation [tex]\( 2^{x-4} = 4 a^{x-6} \)[/tex], follow these detailed steps:
1. Express the constant in terms of the same base:
Given the equation [tex]\( 2^{x-4} = 4 a^{x-6} \)[/tex], note that [tex]\( 4 \)[/tex] can be written as [tex]\( 2^2 \)[/tex]. This allows us to rewrite the equation as:
[tex]\[
2^{x-4} = 2^2 \cdot a^{x-6}
\][/tex]
2. Rewrite the equation emphasizing the exponents:
Now, the equation becomes:
[tex]\[
2^{x-4} = 2^2 \cdot a^{x-6}
\][/tex]
3. Equate the exponents:
To eliminate the powers of 2 and compare the expressions more directly, we can use properties of exponents. Our goal is to equate the exponents involving base 2. We express everything as powers of 2. Notice:
[tex]\[
2^{x-4} = 2^2 \cdot a^{x-6}
\][/tex]
Taking the logarithm base 2 of both sides to simplify:
[tex]\[
\log_2(2^{x-4}) = \log_2(2^2 \cdot a^{x-6})
\][/tex]
4. Simplify the logarithms:
Using the properties of logarithms, we have:
[tex]\[
x-4 = \log_2(2^2 \cdot a^{x-6})
\][/tex]
[tex]\[
x-4 = \log_2(2^2) + \log_2(a^{x-6})
\][/tex]
5. Further simplify using logarithmic identities:
We know that:
[tex]\[
\log_2(2^2) = 2 \quad \text{and} \quad \log_2(a^{x-6}) = (x-6) \log_2(a)
\][/tex]
This gives:
[tex]\[
x-4 = 2 + (x-6) \log_2(a)
\][/tex]
6. Isolate the logarithmic term:
Rearrange to solve for [tex]\( \log_2(a) \)[/tex]:
[tex]\[
x-4 - 2 = (x-6) \log_2(a)
\][/tex]
[tex]\[
(x-4) - 2 = (x-6) \log_2(a)
\][/tex]
[tex]\[
x - 6 = (x-6) \log_2(a)
\][/tex]
7. Divide both sides by [tex]\( x-6 \)[/tex] assuming [tex]\( x \neq 6 \)[/tex]:
We solve the equation [tex]\( x - 6 = (x-6) \log_2(a) \)[/tex], dividing both sides by [tex]\( x-6 \)[/tex]:
[tex]\[
1 = \log_2(a)
\][/tex]
8. Solve for [tex]\( a \)[/tex]:
Using the result from the logarithm:
[tex]\[
a = 2^1 = 2
\][/tex]
Therefore, the value of [tex]\( a \)[/tex] is [tex]\( \boxed{2} \)[/tex].
