Problem 51 - Calculus and its applications

Question

Solve for \(t\). $$ e^{3 t}=900 $$

Step-by-Step Solution

Answer

\(t \approx 2.267\)

Step 1: Understand the Equation

The problem asks us to solve the equation \(e^{3t} = 900\) for \(t\). This is an exponential equation where \(e\) is the base (Euler's number) and \(3t\) is the exponent.

Step 2: Apply the Natural Logarithm

To solve for \(t\), take the natural logarithm (\(\ln\)) of both sides of the equation:\[ \ln(e^{3t}) = \ln(900) \] This step allows us to use the property of logarithms that \(\ln(e^x) = x\).

Step 3: Simplify Using Logarithmic Properties

Utilize the property \(\ln(e^x) = x\) to simplify the left-hand side of the equation:\[ 3t = \ln(900) \] Now, the equation is expressed in terms of \(t\).

Step 4: Isolate \(t\)

To solve for \(t\), divide both sides of the equation by 3:\[ t = \frac{\ln(900)}{3} \]

Step 5: Calculate the Value

Use a calculator to find the natural logarithm of 900 and then perform the division:\(\ln(900) \approx 6.802\), so:\[ t \approx \frac{6.802}{3} \approx 2.267 \]

Key Concepts

Natural Logarithm

The natural logarithm is a special type of logarithm with the base Euler's number, denoted as \(e\). Euler's number \(e\) is approximately 2.71828 and is a fundamental constant in mathematics, similar to pi (\(\pi\)). The natural logarithm is written as \(\ln(x)\), where \(x\) is the number on which the logarithm is being performed. This logarithm is particularly useful in calculus and exponential functions because of its natural relationship with growth processes.
The role of the natural logarithm is to help us simplify expressions where exponential terms are present. For example, when you have an expression like \(e^{3t} = 900\), taking the natural logarithm of both sides allows you to bring the exponent down, transforming \(\ln(e^{3t})\) into \(3t\). This log rule because \(\ln(e^x)\) equals \(x\) is called the inverse property because logarithms are the inverse operations of exponentials.

Logarithmic Properties

Logarithmic properties provide useful tools for manipulating logarithms. One of the most important properties is the power rule: \(\ln(a^b) = b \cdot \ln(a)\). This rule allows you to take the exponent out of a log, simplifying the process of solving equations that contain exponential terms.

• Product Rule: Helps in adding logarithms. Given by: \(\ln(ab) = \ln(a) + \ln(b)\).
• Quotient Rule: Helps in subtracting logarithms. Given by: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\).
• Inverse Property: Useful for canceling out exponents, as \(\ln(e^x) = x\).

In our example, when solving \(e^{3t} = 900\), we apply the inverse property, \(\ln(e^{3t}) = 3t\), making the exponent manageable and allowing us to further isolate \(t\). By understanding these properties, we can effectively solve exponential equations.

Solving Equations

Solving exponentials often involves a series of steps to isolate the desired variable. First, identify the variable you are solving for—in this example, it is \(t\). Recognize that the variable \(t\) is part of an exponent in the form \(e^{3t} = 900\).
To find \(t\), follow these straightforward steps:

• Take the natural logarithm of both sides: Apply \(\ln\) to both sides of the equation: \(\ln(e^{3t}) = \ln(900)\).
• Simplify using properties: Use \(\ln(e^x) = x\), transforming the equation to \(3t = \ln(900)\).
• Isolate the variable: Divide by 3, leading to \(t = \frac{\ln(900)}{3}\).
• Final calculation: Compute \(\ln(900)\) which is approximately 6.802, and divide it by 3 to solve for \(t\), giving \(t \approx 2.267\).

These steps showcase the method to solve for variables in exponential equations using logarithmic functions efficiently.