What is the value of $x$ if $5^{x+2}=5^9$?
A. $x=-11$
B. $x=-7$
C. $x=7$
D. $x=11$
Answer (1)
Recognize that the bases are equal, so the exponents must be equal.
Set the exponents equal to each other: x + 2 = 9 .
Solve for x by subtracting 2 from both sides: x = 9 − 2 .
The value of x is 7 .
Explanation
Understanding the Problem We are given the equation 5 x + 2 = 5 9 . Our goal is to find the value of x that satisfies this equation. Notice that both sides of the equation have the same base, which is 5. This allows us to equate the exponents.
Equating the Exponents Since the bases are the same, we can set the exponents equal to each other: x + 2 = 9 Now, we need to solve for x .
Solving for x To isolate x , we subtract 2 from both sides of the equation: x + 2 − 2 = 9 − 2 x = 7
Final Answer Therefore, the value of x that satisfies the equation 5 x + 2 = 5 9 is 7.
Examples
Exponential equations are useful in modeling various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if a population doubles every year, we can use an exponential equation to predict the population size after a certain number of years. Similarly, in finance, compound interest calculations rely on exponential equations to determine the future value of an investment.
