If [tex]$3^{2 x+1}=3^{x+5}$[/tex], what is the value of x?
A. 2
B. 3
C. 4
D. 6
Answer (1)
Equate the exponents since the bases are equal: 2 x + 1 = x + 5 .
Subtract x from both sides to get x + 1 = 5 .
Subtract 1 from both sides to isolate x : x = 4 .
The value of x is 4 .
Explanation
Equating Exponents We are given the equation 3 2 x + 1 = 3 x + 5 . Since the bases are equal, we can equate the exponents.
Setting up the Equation So, we have 2 x + 1 = x + 5 . Now, we solve for x .
Isolating x Subtract x from both sides: 2 x − x + 1 = x − x + 5 , which simplifies to x + 1 = 5 .
Solving for x Subtract 1 from both sides: x + 1 − 1 = 5 − 1 , which gives us x = 4 .
Final Answer Therefore, the value of x is 4.
Examples
In real-world scenarios, exponential equations like this can model population growth, radioactive decay, or compound interest. For example, if a bacterial population triples every hour, the equation 3 t describes its growth, where t is the time in hours. Solving exponential equations helps predict future population sizes or determine how long it takes for a substance to decay to a certain level. Understanding these concepts is crucial in fields like biology, finance, and environmental science.
