Problem 8 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$5=2 e^{t}$$

Step-by-Step Solution

Answer

\(t = \ln\left(\frac{5}{2}\right)\).

Step 1: Isolate the Exponential Expression

Start by isolating the exponential term from the equation. You currently have \(5 = 2e^t\). To isolate \(e^t\), divide both sides of the equation by 2. This gives us \(\frac{5}{2} = e^t\).

Step 2: Apply the Natural Logarithm

To solve for \(t\), apply the natural logarithm (\(\ln\)) to both sides of the equation. This gives us \(\ln\left(\frac{5}{2}\right) = \ln(e^t)\).

Step 3: Simplify Using Logarithm Properties

Simplify the right side of the equation using the property \(\ln(e^t) = t\). This means our equation becomes \(\ln\left(\frac{5}{2}\right) = t\).

Step 4: Solve for t

We have now solved for \(t\): \(t = \ln\left(\frac{5}{2}\right)\). Use a calculator to evaluate this expression numerically if needed.

Key Concepts

Solving Exponential Equations

Solving exponential equations involves finding the value of the unknown variable that makes the equation true. These equations often contain expressions like \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718.
To solve an exponential equation like \(5 = 2e^t\), the goal is to find the value of \(t\). Unlike linear equations, where we simply "undo" operations, solving exponential equations typically requires the use of logarithms.
Understanding the connection between exponential functions and logarithms is crucial in this process. Utilizing properties of logarithms helps convert the equation into a form that can be easily handled, revealing the solution.

Isolate the Exponential Expression

The first step in solving an exponential equation such as \(5 = 2e^t\) is isolating the exponential expression. This means getting \(e^t\) alone on one side of the equation.
To do this, you divide both sides of the equation by the multiplier of the exponential term. In this case, divide both sides by 2, resulting in \(\frac{5}{2} = e^t\).
When isolating exponential expressions, remember to perform the same operation on both sides of the equation to maintain balance. This step sets us up for applying logarithms in the next part of the process.

Apply Logarithm

After isolating the exponential expression, the next step is to apply the logarithm. For natural exponentials, we use the natural logarithm, denoted as \(\ln\).
Applying \(\ln\) to both sides turns \(\frac{5}{2} = e^t\) into \(\ln\left(\frac{5}{2}\right) = \ln(e^t)\). Applying the logarithm is a crucial move because it allows us to pull down the exponent and solve for the unknown.
This transformation follows one of the fundamental properties of logarithms, which allows the exponent of an exponentiated term to be moved in front of the logarithm.

Simplify Logarithmic Expression

Once you apply the logarithm to both sides, the next step is simplifying the equation. Using properties of logarithms, we know that \(\ln(e^t) = t\). This simplifies the equation to \(\ln\left(\frac{5}{2}\right) = t\).
This step leverages the fact that the natural logarithm and the exponential function \(e\) are inverses. Therefore, \(\ln(e^t)\) simplifies directly to \(t\).
By simplifying, we isolate the variable \(t\), revealing it as \(t = \ln\left(\frac{5}{2}\right)\).

• At this point, you have solved the equation for the variable.
• Finally, you can evaluate \(\ln\left(\frac{5}{2}\right)\) using a calculator to find a numerical answer.