Solve for t.
[tex]e^{-0.24 t}=0.59[/tex]
Answer (2)
To solve for t in the equation e − 0.24 t = 0.59 , take the natural logarithm of both sides to simplify the equation. After isolating t , we calculate its value as approximately 2.199. Therefore, the final answer is t ≈ 2.199 .
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Take the natural logarithm of both sides: ln ( e − 0.24 t ) = ln ( 0.59 ) .
Simplify using logarithm properties: − 0.24 t = ln ( 0.59 ) .
Isolate t by dividing: t = − 0.24 l n ( 0.59 ) .
Calculate the value of t : t ≈ 2.198 .
Explanation
Problem Setup We are given the equation e − 0.24 t = 0.59 and we want to solve for t .
Taking the Natural Logarithm To solve for t , we can take the natural logarithm of both sides of the equation. This gives us: ln ( e − 0.24 t ) = ln ( 0.59 )
Simplifying the Equation Using the property of logarithms, we can simplify the left side of the equation: − 0.24 t = ln ( 0.59 )
Isolating t Now, we can isolate t by dividing both sides of the equation by − 0.24 : t = − 0.24 ln ( 0.59 )
Calculating the Value of t We know that ln ( 0.59 ) ≈ − 0.527632742082372 . Therefore, t = − 0.24 − 0.527632742082372 ≈ 2.19846975867655
Final Answer Therefore, the value of t is approximately 2.198 .
Examples
Exponential decay is a common phenomenon in various fields. For instance, in finance, the value of an asset might decrease exponentially over time. If an asset's value is described by V ( t ) = V 0 e − k t , where V 0 is the initial value, V ( t ) is the value at time t , and k is a constant decay rate, solving for t when V ( t ) reaches a certain level involves solving an equation similar to the one above. This helps investors determine how long it will take for an asset to reach a specific value.
