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 .
Simplify the fraction: x = 7 100 .
The final answer is 7 100 .
Explanation
Problem Setup We are given the equation 500 c d o t 2 0.07 x = 1000 and we want to solve for x .
Isolating the Exponential Term First, we divide both sides of the equation by 500 to isolate the exponential term: 500 500 c d o t 2 0.07 x = 500 1000
Simplifying the Equation Simplifying the equation, we get: 2 0.07 x = 2
Equating the Exponents Since the bases are equal, we can 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
Simplifying the Fraction To simplify the expression, we can multiply the numerator and denominator by 100: x = 7 100
Final Answer Therefore, the solution for x is 7 100 .
Examples
Imagine you are investing money in a bank account that offers a certain interest rate. This problem is similar to calculating how long it takes for your investment to double, given a specific interest rate. Understanding exponential growth and how to solve for the exponent can help you make informed decisions about your investments and financial planning. For example, if you invest $500 and want it to become $1000, and the interest is compounded in a specific way, you can use similar equations to determine the time it takes to reach your goal.
To solve the equation 500 ⋅ 2 0.07 x = 1000 , we first isolate the exponential term to obtain 2 0.07 x = 2 . Then, we equate the exponents and find that x = 7 100 . The final answer is approximately 14.29.
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