Problem 49 - Calculus and its applications

Question

Solve for \(t\). $$ e^{t}=80 $$

Step-by-Step Solution

Answer

The value of \( t \) is approximately 4.3820.

Step 1: Understand the Equation

The given equation is \( e^t = 80 \). This is an exponential equation where \( e \) is the base of the natural logarithms and \( t \) is the exponent we need to solve for.

Step 2: Apply the Natural Logarithm

To isolate \( t \), take the natural logarithm of both sides of the equation. This uses the property that the natural logarithm \( \ln(e^x) = x \). Thus, we have: \( \ln(e^t) = \ln(80) \).

Step 3: Simplify Using Logarithmic Properties

Applying the logarithmic identity from Step 2, \( \ln(e^t) = t \), the equation simplifies to \( t = \ln(80) \).

Step 4: Calculate the Natural Logarithm

Use a calculator to evaluate \( \ln(80) \). You will find that \( t \approx 4.3820 \).

Key Concepts

Natural Logarithms

Natural logarithms are a specific type of logarithm where the base is the constant \( e \). The constant \( e \) is approximately equal to 2.71828. This unique number arises naturally in mathematics, especially in calculations involving growth processes, such as compound interest or population growth.

Natural logarithms are often represented using the notation \( \ln(x) \).

• For natural logarithms, the function \( \ln(e^x) = x \) is particularly significant. This is because applying the natural logarithm to an exponential function with base \( e \) essentially "undoes" the exponential nature, leaving you with the exponent.
• For example, if you have an equation like \( e^t = 80 \), taking the natural logarithm of both sides will give you \( \ln(e^t) = \ln(80) \).

The property \( \ln(e^x) = x \) helps to solve equations where an exponent is present by simplifying the problem to evaluating a logarithm.

Exponential Functions

Exponential functions are a type of mathematical function where a constant base is raised to a variable exponent. The function is typically written as \( f(x) = a^x \) where \( a \) is a positive constant called the base, and \( x \) is the exponent.

Exponential functions with the base \( e \), written as \( f(x) = e^x \), are especially important in many real-world applications.

• Such functions are used to describe continual growth processes, like radioactive decay or population expansion.
• The natural exponential function \( e^x \) has the unique quality that its rate of growth is proportional to its current value, making it useful in calculus and differential equations.

When solving exponential equations, like \( e^t = 80 \), the goal is often to isolate the variable in the exponent (here, \( t \)), which can be effectively done using logarithms.

Logarithmic Properties

Logarithmic properties are essential tools in simplifying and solving equations involving exponential terms. When working with logarithms, certain identities and properties allow us to manipulate and understand the expressions more effectively.

• One of the main properties is \( \ln(e^x) = x \). This comes from the definition of logarithms as the inverse of exponentiation and is key to solving exponential equations.
• Another fundamental property is that \( \ln(ab) = \ln(a) + \ln(b) \). This property allows you to split the logarithm of a product into a sum of logarithms, which can make complex expressions simpler to work with.
• Additionally, \( \ln(a^b) = b \cdot \ln(a) \) helps to pull down exponents, turning a potentially difficult exponential problem into a manageable linear one.

By understanding these properties, you can confidently tackle a wide range of mathematical problems involving exponential and logarithmic functions, just as seen in the solution of the equation \( e^t = 80 \).