Problem 14 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$2 P=P e^{0.3 t}$$

Step-by-Step Solution

Answer

The value of \(t\) is approximately 2.31.

Step 1: Simplify the Equation

Begin by simplifying the equation \(2P = Pe^{0.3t}\). Cancel the \(P\) on both sides of the equation to get \(2 = e^{0.3t}\).

Step 2: Apply the Natural Logarithm

To solve for \(t\), take the natural logarithm (\(\ln\)) of both sides of the equation: \(\ln(2) = \ln(e^{0.3t})\).

Step 3: Use the Logarithm Power Rule

Apply the power rule of logarithms, \(\ln(e^x) = x\), to the right side of the equation: \(\ln(2) = 0.3t\).

Step 4: Solve for t

Divide both sides by 0.3 to isolate \(t\): \(t = \frac{\ln(2)}{0.3}\).

Step 5: Calculate the Result

Calculate the value of \(t\) using the natural logarithm \(\ln(2)\). The approximate value is \(t \approx \frac{0.693}{0.3} \approx 2.31\).

Key Concepts

Exponential Equations

An exponential equation is an equation in which variables appear in exponents. These types of equations are common in many fields, including finance, biology, and physics.

In our original exercise, the equation given is \(2P = Pe^{0.3t}\). In exponential equations like this, the key focus is on the expression \(e^{0.3t}\), where \(e\) is the base of the natural logarithm, approximately equal to 2.718.

Steps like canceling out common factors are essential for simplifying these equations. For instance, removing the factor \(P\) allowed us to simplify the left side to \(2\), setting the pace for solving for \(t\). Remember:

• An equation with a variable in the exponent often requires logarithms for its resolution.
• The base \(e\) signifies exponential growth; understanding its properties is vital for effectively dealing with such equations.

Logarithm Properties

Logarithms and their properties are powerful tools for solving exponential equations. The most commonly used properties include:

• The Power Rule: This states that \(\ln(e^x) = x\). It is a key property because applying the natural logarithm "undoes" exponentiation, effectively bringing variables down from the exponents into more manageable forms.
• Product Rule: This property states that \(\ln(ab) = \ln(a) + \ln(b)\).
• Quotient Rule: According to this rule, \(\ln(a/b) = \ln(a) - \ln(b)\).

In the task at hand, applying \(\ln\) to both sides of the equation \(2 = e^{0.3t}\) enabled us to use the power rule and simplify the exponential expression to just \(0.3t\). This is an example of how logarithm properties, especially the power rule, can simplify complex equations into linear forms for easier solutions.

Solving Equations with Logarithms

Solving exponential equations often relies heavily on using logarithms, which are the inverse operations of exponentials.

The core steps typically include:

• Simplifying the Equation: As shown, simplifying \(2P = Pe^{0.3t}\) to \(2 = e^{0.3t}\) is a crucial first step.
• Applying Logarithms: Taking the natural logarithm of both sides, \(\ln(2) = \ln(e^{0.3t})\), utilizes the inverse nature of logarithms to bring variables out of the exponent.
• Using Logarithm Properties: Applying the power rule, you simplify the equation further to \(0.3t = \ln(2)\).
• Isolating the Variable: Solving for \(t\) by dividing both sides by 0.3 gives \(t = \frac{\ln(2)}{0.3}\).
• Calculating the Result: Use a calculator to find \(\ln(2)\) which is about 0.693, then divide by 0.3 to find \(t \approx 2.31\).

This process illustrates how logarithms help in transitioning the variable from an exponent to being isolated, aiding in finding a solution efficiently.