Let's solve the equation [tex]\( 4^{x-1} = \log_8 64 \)[/tex] step-by-step.
1. Evaluate the logarithm:
We start by solving [tex]\( \log_8 64 \)[/tex]. This asks: to what power must 8 be raised to get 64?
Notice that:
[tex]\[
8^2 = 64
\][/tex]
Therefore:
[tex]\[
\log_8 64 = 2
\][/tex]
2. Substitute the logarithm back into the equation:
With [tex]\( \log_8 64 = 2 \)[/tex], the original equation [tex]\( 4^{x-1} = \log_8 64 \)[/tex] simplifies to:
[tex]\[
4^{x-1} = 2
\][/tex]
3. Rewrite the equation with a common base:
Next, it helps to express both sides of the equation using the same base. We know that:
[tex]\[
4 = 2^2
\][/tex]
So:
[tex]\[
4^{x-1} = (2^2)^{x-1} = 2^{2(x-1)}
\][/tex]
Now our equation looks like this:
[tex]\[
2^{2(x-1)} = 2^1
\][/tex]
4. Set the exponents equal to each other:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
2(x-1) = 1
\][/tex]
5. Solve for [tex]\( x \)[/tex]:
Now, we solve the equation for [tex]\( x \)[/tex]:
[tex]\[
2(x-1) = 1
\][/tex]
[tex]\[
2x - 2 = 1
\][/tex]
[tex]\[
2x = 3
\][/tex]
[tex]\[
x = \frac{3}{2}
\][/tex]
Thus, the solution to the equation [tex]\( 4^{x-1} = \log_8 64 \)[/tex] is [tex]\( x = \frac{3}{2} \)[/tex], which corresponds to option (B).
So, the correct answer is:
[tex]\[
\boxed{\frac{3}{2}}
\][/tex]
