To solve the equation [tex]\(2^{x-4} = 4 b^{x-6}\)[/tex] for [tex]\(b\)[/tex], we need to follow several steps to simplify and analyze the equation.
### Step 1: Simplify the Equation
Given the equation:
[tex]\[ 2^{x-4} = 4 b^{x-6} \][/tex]
Recall that 4 can be expressed as [tex]\(2^2\)[/tex]. Therefore, we can rewrite the equation as:
[tex]\[ 2^{x-4} = 2^2 \cdot b^{x-6} \][/tex]
### Step 2: Use Properties of Exponents
We have the term [tex]\(2^2 \cdot b^{x-6}\)[/tex]. Let's denote [tex]\(4\)[/tex] as [tex]\(2^2\)[/tex]:
[tex]\[ 2^{x-4} = 2^2 \cdot b^{x-6} \][/tex]
To simplify further, isolate the base 2 terms:
[tex]\[ 2^{x-4} = 2^2 \cdot b^{x-6} \][/tex]
### Step 3: Rewrite in Logarithmic Form
Next, take the natural logarithm or recognize that since the bases are the same, the exponents must be equal if the bases are consistent. By equating the exponents on both sides, we can solve for [tex]\(b\)[/tex]:
[tex]\[ x - 4 = 2 + \log_b(b^{x-6}) \][/tex]
Since [tex]\(\log_b(b^{x-6}) = x-6\)[/tex], the equation simplifies to:
[tex]\[ x-4 = 2 + (x-6) \][/tex]
### Step 4: Solve the Simplified Equation
Now, solve for [tex]\(b\)[/tex] by isolating it. Begin by canceling [tex]\(x\)[/tex] from both sides:
[tex]\[ x - 4 = 2 + x - 6 \][/tex]
[tex]\[ x - 4 = x - 4 \][/tex]
Here we recognize that no values of [tex]\(b\)[/tex] in the given options satisfy this derived equation, meaning there are no real, consistent values of [tex]\(b\)[/tex] among the provided possibilities.
### Conclusion
Analyzing the equation and evaluating the possible values given in the choices (6, 2, 8), we find no value that fits the equation consistently.
Therefore, the correct response is:
[tex]\[ b \][/tex] has no valid solution among the provided options.
Hence, the solution set is empty, indicating there is no valid value for [tex]\(b\)[/tex].
