Problem 55 - Calculus and its applications

Question

Solve for \(t\). $$ e^{-0.02 t}=0.06 $$

Step-by-Step Solution

Answer

\(t = \frac{\ln(0.06)}{-0.02}\), calculate using a calculator.

Step 1: Remove the exponential

To remove the exponential, take the natural logarithm (ln) of both sides of the equation. The equation becomes:\[\ln(e^{-0.02 t}) = \ln(0.06)\]

Step 2: Simplify the left side

Using the property of logarithms, \( \ln(e^x) = x \), simplify the left side of the equation:\[-0.02 t = \ln(0.06)\]

Step 3: Solve for \(t\)

Divide both sides of the equation by \(-0.02\) to solve for \(t\):\[t = \frac{\ln(0.06)}{-0.02}\] Calculate \( \ln(0.06) \) and solve for \(t\). Use a calculator for the logarithm to find the final value.

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special kind of logarithm with base \( e \), where \( e \) is approximately 2.71828. It's often used when dealing with exponential equations or when we need to reverse the effect of exponentiation with \( e \). The beauty of the natural logarithm lies in its simplicity—if you have an equation with \( e^x \), applying \( \ln \) makes it straightforward to solve for \( x \). In general terms, if \( \ln(y) = x \), then \( e^x = y \).

• The natural logarithm is the inverse of the exponential function. This means that \( \ln(e^x) = x \).
• It is particularly useful for solving equations where the variable is an exponent.

When solving exponential equations, applying the natural logarithm helps you "bring down" the exponent as a factor, making it easier to isolate the variable and solve.

Key Logarithmic Properties to Simplify Solutions

Logarithmic properties are essential tools when working with equations involving logarithms. They let you transform and simplify calculations that involve logarithms, making it easier to solve equations. Here are a few critical properties of logarithms you should remember:

• 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 our specific case, we used \( \ln(e^{-0.02 t}) = -0.02 t \), a clear example of the Power Property, which allows us to bring the exponent down as a coefficient.

These properties enable us to handle complex expressions, leading us seamlessly into the problem-solving process.

Solving Exponential Equations

To solve exponential equations, such as \( e^{-0.02 t} = 0.06 \), we typically follow a series of steps that utilize both the natural logarithm and its properties. Here’s a step-by-step breakdown:

1. **Apply the Natural Logarithm:** - By taking the natural logarithm on both sides, we can deal with the exponent directly. Our equation turns into \( \ln(e^{-0.02 t}) = \ln(0.06) \).2. **Use Logarithmic Properties:** - Simplify using the property \( \ln(e^x) = x \). It gives us \( -0.02 t = \ln(0.06) \). 3. **Isolate the Variable:** - To find \( t \), divide each side by \(-0.02\). This results in the equation \( t = \frac{\ln(0.06)}{-0.02} \).4. **Calculate the Solution:** - Use a calculator to find \( \ln(0.06) \), then divide by \(-0.02\) to determine the value of \( t \).By mastering these steps, you can solve any exponential equation that involves the natural logarithm with confidence. Each step logically follows from the previous, providing a clear path to the solution.