Problem 9 - Applied Calculus

Question

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$e^{3 t}=100$$

Step-by-Step Solution

Answer

The solution for \(t\) is approximately 1.535.

Step 1: Understand the Equation

The equation given is an exponential equation: \(e^{3t} = 100\). Our goal is to solve for the variable \(t\).

Step 2: Apply the Natural Logarithm

To solve for \(t\), apply the natural logarithm (ln) to both sides of the equation to eliminate the exponential function. This gives us: \(\ln(e^{3t}) = \ln(100)\).

Step 3: Simplify the Left Side Using Logarithm Rules

The property of logarithms allows us to bring down exponents as multipliers. So, \(\ln(e^{3t})\) simplifies to \(3t \cdot \ln(e)\). Since \(\ln(e) = 1\), this further simplifies to \(3t\).

Step 4: Solve for t

Now, the equation is \(3t = \ln(100)\). Divide both sides by 3 to isolate \(t\): \(t = \frac{\ln(100)}{3}\).

Step 5: Calculate the Result

Using a calculator, find \(\ln(100)\) and divide by 3 to get the numerical value for \(t\). The approximate value of \(\ln(100)\) is about 4.605, so \(t \approx \frac{4.605}{3} \approx 1.535\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent. They are in the form \( a^{x} = b \), where \( a \) is the base and \( b \) is a constant. In our exercise, the base is the natural number \( e \), approximately equal to 2.718, which is unique to exponential functions involving natural logarithms.
Exponential equations model growth processes, such as population growth or compound interest. These are equations where the unknown variable appears as an exponent. To solve an exponential equation, such as \( e^{3t} = 100 \), we often use logarithms, which allow us to handle the exponent directly.
Recognizing the form of an exponential equation is important because it guides you in deciding to use logarithmic operations to solve it. The next step is choosing the right type of logarithm—natural logarithm, in this case, because it aligns with the base \( e \).

Logarithm Properties

Logarithms are the inverse operations of exponentials. They help convert multiplication into addition, simplifying the process of solving equations involving exponents. The natural logarithm, denoted as \( \ln \), specifically uses the base of \( e \).
There are several key properties of logarithms that assist in solving equations:

• **Product Property**: \( \ln(ab) = \ln(a) + \ln(b) \)
• **Quotient Property**: \( \ln\left( \frac{a}{b} \right) = \ln(a) - \ln(b) \)
• **Power Property**: \( \ln(a^b) = b \cdot \ln(a) \)

In the original problem, the Power Property is crucial. When we take \( \ln(e^{3t}) \), it simplifies to \( 3t \cdot \ln(e) \). Since \( \ln(e) = 1 \), this results in \( 3t \). Knowing these properties helps in breaking down complex logarithmic expressions into manageable calculations.

Solving Equations with Logarithms

To solve an equation like \( e^{3t} = 100 \), the application of logarithms is a convenient method. Taking the logarithm of both sides simplifies the process by removing the exponent.

Here is a simple step-by-step approach:

• **Apply the natural logarithm:** Convert the exponential form using logarithms. If \( e^{3t} = 100 \), then \( \ln(e^{3t}) = \ln(100) \).
• **Simplify using logarithm rules:** The equation \( \ln(e^{3t}) \) becomes \( 3t \cdot \ln(e) \). Because \( \ln(e) \) is equal to 1, you have \( 3t = \ln(100) \).
• **Isolate the variable**: Divide both sides by 3 to solve for \( t \). This gives \( t = \frac{\ln(100)}{3} \).

By following these steps and using a calculator, you can find that \( \ln(100) \) is approximately 4.605. Thus, \( t \approx \frac{4.605}{3} \approx 1.535 \). This method underlines the usefulness of logarithms in solving exponential equations, breaking down seemingly complex formulas into simple arithmetic tasks.