Problem 54 - Calculus and its applications

Question

Solve for \(t\). $$ e^{-t}=0.1 $$

Step-by-Step Solution

Answer

t \approx 2.302

Step 1: Understand the Equation

The equation given is \( e^{-t} = 0.1 \). This is an exponential equation where the left side is an exponential function with base \( e \) and exponent \( -t \), and the right side is a constant, 0.1.

Step 2: Take the Natural Logarithm of Both Sides

To solve for \( t \), we can use logarithms to eliminate the exponential. Taking the natural logarithm (\( \ln \)) of both sides, we get:\[ \ln(e^{-t}) = \ln(0.1) \]

Step 3: Apply Logarithm Power Rule

Using the power rule of logarithms, \( \ln(e^a) = a \cdot \ln(e) = a \), we simplify the left side to:\[ -t = \ln(0.1) \]

Step 4: Solve for t

To isolate \( t \), multiply both sides of the equation by -1:\[ t = -\ln(0.1) \]

Step 5: Evaluate the Natural Logarithm

Use a calculator to find \( \ln(0.1) \). This value is approximately -2.302. Thus, multiplying by -1 gives:\[ t \approx 2.302 \]

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is a special type of logarithm with the base \( e \), an irrational constant approximately equal to 2.71828. It is commonly used in calculus and mathematical modeling because of its nice properties, especially when dealing with growth processes. The natural log helps in simplifying exponential expressions and solving equations like the one in our exercise. When you take the natural logarithm of an exponential function having base \( e \), you can easily solve for the variable.

Here's how it works:

• The expression \( \ln(e^x) \) simplifies to \( x \) because \( \ln(e) = 1 \).
• Applying \( \ln \) on both sides of an exponential equation helps "bring down" the exponent, making it more accessible for further simplification.

In the exercise, using the natural logarithm was pivotal in solving for \( t \), as it transformed \( e^{-t} \) into a manageable equation.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. The general form is \( a^x \), where \( a \) is the base and \( x \) is the exponent. In our exercise, the base is \( e \), and \( e^{-t} \) is the exponential term.

Exponential functions are known for their rapid growth or decay:

• **Growth:** When the exponent is positive, the function shows exponential growth, increasing rapidly.
• **Decay:** When the exponent, like in our case with \(-t\), is negative, the function represents exponential decay, decreasing as the variable increases.

Understanding exponential functions is crucial, as they model many natural phenomena, including population growth and radioactive decay. The ability to manipulate these functions using tools like logarithms, as shown in the problem, is a valuable skill in STEM fields.

Logarithmic Properties

Logarithms are the inverse operations of exponential functions, serving as a way to solve equations involving exponentials. Some properties of logarithms simplify computations and problem-solving:

• **Power Rule:** \( \ln(a^b) = b \cdot \ln(a) \) allows you to bring the exponent down. This property was used in our solution to transform \( \ln(e^{-t}) \) into \(-t\) after recognizing that \( \ln(e) = 1 \).
• **Product Rule:** \( \ln(ab) = \ln(a) + \ln(b) \) separates products inside a log into the sum of logs.
• **Quotient Rule:** \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \) breaks down the division inside a log into the difference of logs.

Such properties are extremely useful in transforming and solving equations involving logarithms, allowing more straightforward algebraic manipulation. Mastery of these properties can greatly enhance your problem-solving toolkit, particularly in finance and scientific computations.