Solve for $x$.
$2000=500 e^{4 x}$
A. $x=\ln (4)-4$
B. $x=\frac{1}{e}$
C. $x=\frac{\ln (4)}{4}$
D. $x=\frac{\ln (2000)}{4 \cdot \ln (500)}$
Answer (2)
Divide both sides of the equation by 500: 4 = e 4 x .
Take the natural logarithm of both sides: ln ( 4 ) = 4 x .
Divide by 4 to isolate x : x = 4 l n ( 4 ) .
The solution is 4 ln ( 4 ) .
Explanation
Problem Analysis We are given the equation 2000 = 500 e 4 x and asked to solve for x . This involves isolating x by using algebraic manipulations and properties of logarithms.
Isolating the Exponential Term First, divide both sides of the equation by 500 to isolate the exponential term: 500 2000 = 500 500 e 4 x 4 = e 4 x
Applying Natural Logarithm Next, take the natural logarithm of both sides of the equation to remove the exponential: ln ( 4 ) = ln ( e 4 x ) Using the property of logarithms, ln ( e a ) = a , we get: ln ( 4 ) = 4 x
Solving for x Now, divide both sides by 4 to solve for x :
x = 4 ln ( 4 )
Final Answer Therefore, the solution for x is 4 l n ( 4 ) .
Examples
Exponential equations like the one we solved are used in various fields such as finance, biology, and physics. For example, they can model population growth, radioactive decay, or the accumulation of interest in a bank account. Understanding how to solve these equations allows us to predict future values or determine past conditions in these scenarios. For instance, if we know the rate at which a population is growing, we can use an exponential equation to estimate the population size at a future time.
The solution to the equation 2000 = 500 e 4 x is x = 4 l n ( 4 ) , which is option C. This was found by isolating the exponential term, taking the natural logarithm, and solving for x .
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