Solve the equation.
[tex]$\begin{array}{c}
500 \cdot 2^{0.07 x}=1000 \\
x=[?]\\
\end{array}$[/tex]
Answer (2)
Divide both sides of the equation by 500: 2 0.07 x = 2 .
Equate the exponents: 0.07 x = 1 .
Solve for x : x = 0.07 1 .
Calculate the value of x : x = 7 100 .
The solution to the equation is 7 100 .
Explanation
Problem Analysis We are given the equation 500 c d o t 2 0.07 x = 1000 and we need to solve for x .
Isolating the Exponential Term First, we divide both sides of the equation by 500 to isolate the exponential term: 2 0.07 x = 500 1000
Simplifying the Equation Next, we simplify the right side of the equation: 2 0.07 x = 2
Equating the Exponents Since the bases are equal, we equate the exponents: 0.07 x = 1
Solving for x Now, we solve for x by dividing both sides by 0.07: x = 0.07 1
Calculating the Value of x Finally, we calculate the value of x : x = 0.07 1 = 7 100 ≈ 14.2857
Final Answer Thus, the solution to the equation is x = 7 100 .
Examples
Exponential equations are used in various real-world applications, such as modeling population growth, radioactive decay, and compound interest. For instance, if you invest money in a bank account with a fixed interest rate compounded continuously, the amount of money you have after a certain time can be modeled using an exponential equation. Solving such equations helps you determine how long it will take for your investment to reach a specific target amount. Understanding exponential growth and decay is crucial in finance, biology, and physics.
To solve the equation 500 ⋅ 2 0.07 x = 1000 , we isolate the exponential term and find that x = 7 100 . This value can also be approximated as 14.29.
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