Problem 17 - Introductory and Intermediate Algebra for College Students 4th

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 to the equation \(4^{x}=\frac{1}{\sqrt{2}}\) is \(x = -\frac{1}{4}\)

Step 1: Converting to the same base

Express both sides of the equation as a power of 2. Since 4 is \(2^2\), write \(4^{x}\) as \((2^2)^x\). The right side expression, \(\frac{1}{\sqrt{2}}\), can also be written as \(2^{-\frac{1}{2}}\). This leads to \((2^2)^x = 2^{-\frac{1}{2}}\).

Step 2: Simplify the power expression

Next, simplify the left side of the expression using the power of power rule, which says that \((a^m)^n = a^{mn}\). So, we have \(2^{2x} = 2^{-\frac{1}{2}}\). Now both sides of the equation are expressed as a power of 2.

Step 3: Equating the Exponents

Since both sides have the same base, the exponents must also be equal to each other. Set the exponents equal to each other and solve for x, yielding \(2x = -\frac{1}{2}\). Solving for x gives us \(x = -\frac{1}{4}\).