Problem 10 - Precalculus : Building Concepts and Connections

Question

Use \(f(t)=10 e^{-t}\). For what value of \(t\) will \(f(t)=2 ?\)

Step-by-Step Solution

Answer

The value of \(t\) for which \(f(t)=2\) is \(t = - \ln(0.2)\)

Step 1: Set the Function Equal to 2

The first thing to do is to set the function \( f(t) = 10e^{-t}\) equal to 2. This gives us the equation \( 10e^{-t} = 2 \).

Step 2: Isolate the Exponential Term

Next, we isolate the exponential term on one side of the equation. This is done by dividing both sides of the equation by 10 to get \( e^{-t} = 0.2 \) .

Step 3: Apply Natural Logarithm

We then apply the natural logarithm to both sides of the equation. The natural logarithm of \( e^{-t} \) is just \(-t\). So, by applying the natural logarithm to both sides, we get the equation \( -t = \ln(0.2) \) .

Step 4: Solve for t

The final step is to solve for \(t\). Because the equation is \(-t = \ln(0.2)\), we multiply both side of the equation by -1 to find \( t = -\ln(0.2)\) .

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a mathematical function that is the inverse of the exponential function \(e^x\), where \(e\) is Euler's number, approximately equal to 2.71828. This means that if \(y = e^x\), then \(x\) is the natural logarithm of \(y\), or \(x = \ln(y)\).