If [tex]$2.8^3=2^{x-4}$[/tex], find the value of x.
Answer (2)
Rewrite 2.8 as a fraction: 2.8 = 5 14 , so the equation becomes ( 5 14 ) 3 = 2 x − 4 .
Take the logarithm base 2 of both sides: lo g 2 (( 5 14 ) 3 ) = lo g 2 ( 2 x − 4 ) .
Simplify using logarithm properties: 3 lo g 2 ( 5 14 ) = x − 4 .
Solve for x : x = 3 lo g 2 ( 5 14 ) + 4 ≈ 8.356 , which rounds to 8 .
Explanation
Problem Analysis We are given the equation 2. 8 3 = 2 x − 4 and asked to find the value of x .
Rewriting the Equation First, let's rewrite 2.8 as a fraction: 2.8 = 10 28 = 5 14 . So the equation becomes ( 5 14 ) 3 = 2 x − 4 .
Taking Logarithms Now, take the logarithm base 2 of both sides of the equation: lo g 2 (( 5 14 ) 3 ) = lo g 2 ( 2 x − 4 ) .
Simplifying the Equation Using logarithm properties, we can simplify the equation: 3 lo g 2 ( 5 14 ) = x − 4 .
Solving for x Now, solve for x : x = 3 lo g 2 ( 5 14 ) + 4 .
Calculating the Value of x Using a calculator, we find that lo g 2 ( 5 14 ) ≈ 1.452093493836908 . Therefore, x ≈ 3 ( 1.452093493836908 ) + 4 ≈ 4.356280481510724 + 4 ≈ 8.356280481510724 .
Final Answer Rounding to the nearest whole number, we get x ≈ 8 .
Examples
Understanding exponential equations is crucial in various fields like finance, where it helps calculate compound interest. For instance, if an investment doubles every 5 years, you can use exponential equations to determine how long it will take to reach a specific financial goal. Similarly, in biology, exponential equations model population growth, allowing scientists to predict how quickly a population might increase or decrease under certain conditions. These equations are also fundamental in physics, particularly in understanding radioactive decay, where they help determine the half-life of radioactive substances. In our problem, we solved for x in the equation 2. 8 3 = 2 x − 4 . We rewrote 2.8 as 5 14 , took the logarithm base 2 of both sides, and simplified to find x = 3 lo g 2 ( 5 14 ) + 4 . This type of problem-solving is essential for modeling and predicting outcomes in various real-world scenarios.
To solve the equation 2. 8 3 = 2 x − 4 , rewrite 2.8 as 5 14 , then take logarithms on both sides leading to x = 3 lo g 2 ( 5 14 ) + 4 . By calculating, we find that x is approximately 8 .
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