b. \( 8^{x-1}=2^{4 x-6} \)

To solve the equation \( 8^{x-1}=2^{4 x-6} \), first, let's express \( 8 \) as a power of \( 2 \). Since \( 8 = 2^3 \), we can rewrite the left side: \[ (2^3)^{x-1} = 2^{4x-6} \] This simplifies to: \[ 2^{3(x-1)} = 2^{4x-6} \] Now that both sides have the same base, we can set the exponents equal to each other: \[ 3(x-1) = 4x - 6 \] Expanding and rearranging gives us: \[ 3x - 3 = 4x - 6 \] \[ -3 + 6 = 4x - 3x \] \[ 3 = x \] Thus, the solution is \( x = 3 \). Happy solving! Remember, transforming bases often makes the problem a whole lot easier!