Questions: Solve the exponential equation by expressing each side as a power of the same base and then equating exponents. 4^x = 64 The solution set is □

Solution Steps

First, we need to express both sides of the equation \(4^x = 64\) as powers of the same base. We know that:

\[ 4 = 2^2 \]

and

\[ 64 = 2^6 \]

Substitute the expressions for 4 and 64 into the original equation:

\[ (2^2)^x = 2^6 \]

Apply the power of a power property \((a^m)^n = a^{m \cdot n}\):

\[ 2^{2x} = 2^6 \]

Since the bases are the same, we can equate the exponents:

\[ 2x = 6 \]

Divide both sides by 2 to solve for \(x\):

\[ x = \frac{6}{2} = 3 \]

Final Answer