Express both bases as powers of the same prime number.
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here is the solution to the indices problem:
2. Find the value of $x$ which satisfies the equation: $16^x = 8^{x-1}$
Step 1: Express both bases as powers of the same prime number.
$$16 = 2^4$$
$$8 = 2^3$$
Step 2: Substitute these into the equation.
$$(2^4)^x = (2^3)^{x-1}$$
Step 3: Apply the power rule $(a^m)^n = a^{mn}$.
$$2^{4x} = 2^{3(x-1)}$$
$$2^{4x} = 2^{3x-3}$$
Step 4: Equate the exponents since the bases are the same.
$$4x = 3x - 3$$
Step 5: Solve for $x$.
$$4x - 3x = -3$$
$$x = -3$$
The value of $x$ is $\boxed{x = -3}$.
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