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Chapter 4: Problem 13

In Exercises \(11-14,\) use \(f(t)=4 e^{t}\) For what value of \(t\) will \(f(t)=8 ?\)

Short Answer

Expert verified
The value of \(t\) for which \(f(t)=8\) is \(t = \ln(2)\).

Step by step solution

01

Set the Function Equal to the Desired Value

To find at what point the function equals 8, write the equation \(4 e^{t} = 8\).
02

Simplify the equation

Solve the equation for \(t\). First, divide each side by 4 to get \(e^{t} = 2\).
03

Solve for \(t\)

Use the natural logarithm formula to solve for \(t\). That results in \(t = \ln(2)\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithms
Natural logarithms are a special type of logarithm with the base of the exponential constant, denoted as "e." The value of "e" is approximately 2.71828. Natural logarithms are often written as \(\ln(x)\) and represent the power to which "e" must be raised to yield the number \(x\). For instance, \(\ln(e) = 1\) because \(e^1 = e\). They are widely used in calculus and solve exponential equations, like the one in our exercise. In summary, natural logarithms help in simplifying complex exponential equations by transforming them into linear formats.
  • Base "e" is different from other bases like 10 in common logarithms.
  • \(\ln(e^x) = x\), showing a natural log's inversive property over exponential functions with the same base.
  • Natural logarithms are crucial for growth models, decay calculations, and in many real-world scientific and engineering problems.
Understanding how to manipulate and calculate using natural logarithms will make solving exponential equations a breeze.
Equation Solving
One of the essential skills in mathematics is solving equations, especially when they contain exponential functions. Exponential equations often involve terms with a base of "e" raised to a power, as in \(f(t) = 4e^t\). Solving these equations involves isolating the variable of interest, typically using inverse operations.

In the provided solution, the equation \(4e^t = 8\) is given. The process to solve this begins with:
  • Divide both sides by 4: Simplifying the equation by getting rid of any multipliers on the "e" term results in \(e^t = 2\).
  • Use the inverse operation: Applying the natural logarithm, \(\ln\), to both sides leads to \(t = \ln(2)\), because \(\ln(e^t) = t\).
By using inverse functions and careful steps, exponential equations can be made simpler to solve. The key is to break down each step to isolate the variable efficiently.
Function Evaluation
Function evaluation is all about plugging values into a given function to get specific outcomes or understand their behavior under designated conditions. In this exercise, we determined for what value of \(t\) the function \(f(t) = 4e^t\) equals 8. The process is straightforward once you grasp function concepts.

Starting with the given function, set it equal to the desired value: \(4e^t = 8\). Each function value corresponds to a specific input, and solving means finding the input that accurately results in the given output.
  • Identify the function form: Recognize that \(f(t)\) is based on an exponential function with a constant factor.
  • Plug and manipulate: Set the function to the goal value and solve it, as shown in our example, to modify values or conditions given in the function.
Recognizing and correctly inputting values into functions provides insight into their behavior, helping us predict outcomes efficiently.

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