Question

solve the equation $8 \cdot 2^x = 4^x$. if no solution exists, enter none.

Explanation:

Step1: Express all terms with base 2

We know that \(8 = 2^3\) and \(4 = 2^2\), so we can rewrite the equation \(8\cdot2^{x}=4^{x}\) as \(2^{3}\cdot2^{x}=(2^{2})^{x}\). Using the exponent rule \(a^{m}\cdot a^{n}=a^{m + n}\) and \((a^{m})^{n}=a^{mn}\), the equation becomes \(2^{x + 3}=2^{2x}\).

Step2: Set the exponents equal

Since the bases are the same and the exponential function \(y = 2^{x}\) is one - to - one (if \(a^{m}=a^{n}\), then \(m = n\) for \(a>0,a
eq1\)), we can set the exponents equal to each other: \(x + 3=2x\).

Step3: Solve for x

Subtract \(x\) from both sides of the equation \(x + 3=2x\). We get \(3=2x - x\), which simplifies to \(x = 3\).

Answer:

\(x = 3\)