Solve for $x$. $2^{3 x-5}=4$ Select the correct choice below and, if necessary, fill in the answer box to complete your A. The solution is $x=$ $\square$ . (Type an integer or a simplified fraction. Use a comma to separate answers as ne B. The solution is not a real number.
Answer (2)
Rewrite the equation with the same base: 2 3 x − 5 = 2 2 .
Equate the exponents: 3 x − 5 = 2 .
Solve for x : 3 x = 7 , so x = 3 7 .
The solution is 3 7 .
Explanation
Problem Analysis We are given the equation 2 3 x − 5 = 4 and we need to solve for x .
Rewriting the Equation First, we rewrite the equation such that both sides have the same base. Since 4 = 2 2 , we can rewrite the equation as 2 3 x − 5 = 2 2 .
Equating the Exponents Since the bases are equal, we can equate the exponents: 3 x − 5 = 2 .
Solving for x Now, we solve the linear equation for x . Add 5 to both sides: 3 x = 2 + 5 , which simplifies to 3 x = 7 .
Isolating x Divide both sides by 3 to isolate x : x = 3 7 .
Checking the Solution To check our solution, we substitute x = 3 7 back into the original equation: 2 3 ( 3 7 ) − 5 = 2 7 − 5 = 2 2 = 4 . The solution is valid.
Final Answer Therefore, the solution is x = 3 7 .
Examples
In real-world scenarios, exponential equations like this can model various phenomena, such as the growth of bacteria or the decay of radioactive substances. For instance, if we know that the population of a bacteria colony doubles every certain period, we can use an exponential equation to determine how long it will take for the population to reach a specific size. By solving for the variable representing time, we can make predictions about the future state of the system. Understanding exponential equations is crucial in fields like biology, physics, and finance for modeling and predicting growth and decay processes.
The solution to the equation 2 3 x − 5 = 4 is x = 3 7 . This is found by rewriting the equation with the same base, equating the exponents, and solving for x . Finally, verifying the solution confirms its correctness.
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