Problem 48 - College Algebra Essentials

Question

Solve each exponential equation. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$2^{2 x}+2^{x}-12=0$$

Step-by-Step Solution

Answer

The solution to the equation \(2^{2x} + 2^x - 12 = 0\) in terms of natural logarithms is \(x = ln(3) \over ln(2)\). The decimal approximation, correct to two decimal places, for \(x\) is 1.58.

Step 1: Rewrite the Equation

Rewrite the equation to identify it as an equivalent quadratic equation. Let \(u = 2^x\). So, the equation \(2^{2x} + 2^{x} - 12 = 0\) becomes \(u^2 + u - 12 = 0\).

Step 2: Solve the Quadratic Equation

Solve the quadratic equation \(u^2 + u - 12 = 0\) by factoring or using the quadratic formula \(-b \pm sqrt{b^2 - 4ac} \over 2a\). Factoring the equation gives us \((u-3)(u+4)=0\). So, \(u = 3\) or \(u = -4\).

Step 3: Substitute Back and Solve the Exponential Equations

Substitute back \(2^x\) for \(u\). This gives us \(2^x = 3\) and \(2^x = -4\). Solve these equations for \(x\). The equation \(2^x = -4\) has no solution because \(2^x\) is always greater than zero. As for the equation \(2^x = 3\), taking the natural logarithm of both sides gives us \(x = ln(3) \over ln(2)\).

Step 4: Calculate Decimal Approximation

Finally, calculate the decimal approximation for the solution using a calculator. Make sure to round the answer off to two decimal places.