Problem 17 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$4^{x}=\frac{1}{\sqrt{2}}$$

Step-by-Step Solution

Answer

The solution is \(x = -\frac{1}{4}\).

Step 1: Rewrite the Left Expression

Rewrite \(4^{x}\) as \((2^{2})^{x}\). This simplifies to \(2^{2x}\). So, the equation becomes \(2^{2x} = \frac{1}{\sqrt{2}}\).

Step 2: Rewrite the Right Expression

Next, rewrite \(\frac{1}{\sqrt{2}}\) as \(2^{-\frac{1}{2}}\). Hence, the equation becomes \(2^{2x} = 2^{-\frac{1}{2}}\). Given that now both sides of the equation have been transformed to have the same base (2), we can equate the exponents.

Step 3: Equate Exponents and Solve for \(x\)

By equating exponents, \(2x = -\frac{1}{2}\). Solving for \(x\) results in \(x = -\frac{1}{4}\).