What is the solution to the equation $2^{x+4}-12=20$?
Answer (1)
Add 12 to both sides of the equation: 2 x + 4 = 32 .
Express 32 as a power of 2: 2 x + 4 = 2 5 .
Equate the exponents: x + 4 = 5 .
Solve for x: x = 1 . The solution to the equation is 1 .
Explanation
Understanding the Problem We are given the equation 2 x + 4 − 12 = 20 . Our goal is to find the value of x that satisfies this equation from the given options.
Isolating the Exponential Term First, we want to isolate the term with the exponent. To do this, we add 12 to both sides of the equation: 2 x + 4 − 12 + 12 = 20 + 12 2 x + 4 = 32
Expressing 32 as a Power of 2 Now, we express 32 as a power of 2. We know that 32 = 2 × 2 × 2 × 2 × 2 = 2 5 . So, we can rewrite the equation as: 2 x + 4 = 2 5
Equating the Exponents Since the bases are equal (both are 2), the exponents must be equal. Therefore, we set the exponents equal to each other: x + 4 = 5
Solving for x To solve for x , we subtract 4 from both sides of the equation: x + 4 − 4 = 5 − 4 x = 1
Final Answer Therefore, the solution to the equation 2 x + 4 − 12 = 20 is x = 1 .
Examples
Exponential equations are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For example, if a bacteria population doubles every hour, we can use an exponential equation to predict the population size after a certain number of hours. Similarly, in finance, compound interest calculations rely on exponential growth to determine the future value of an investment.
