Problem 6 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$100=25(1.5)^{t}$$

Step-by-Step Solution

Answer

The value of \(t\) is approximately 3.42.

Step 1: Isolate the Exponential Expression

Begin by dividing both sides of the equation by 25 to isolate the exponential expression: \[ \frac{100}{25} = (1.5)^{t} \] This simplifies to: \[ 4 = (1.5)^{t} \]

Step 2: Apply the Natural Logarithm

Take the natural logarithm of both sides of the equation to remove the exponent: \[ \ln(4) = \ln((1.5)^{t}) \] By logarithm properties, this simplifies to: \[ \ln(4) = t \cdot \ln(1.5) \]

Step 3: Solve for t

To solve for \(t\), divide both sides by \(\ln(1.5)\):\[ t = \frac{\ln(4)}{\ln(1.5)} \]

Step 4: Calculate t

Using a calculator, compute the values: \[ \ln(4) \approx 1.386 \, \text{and} \, \ln(1.5) \approx 0.405 \] Thus, \[ t \approx \frac{1.386}{0.405} \approx 3.42 \]

Key Concepts

Natural Logarithms

Natural logarithms, denoted as \( \ln \), are a special type of logarithm. They are based on the mathematical constant \( e \), which is approximately 2.718. When we use natural logarithms, we are essentially considering exponents that have \( e \) as their base.
This is particularly useful because many natural phenomena exhibit growth rates that can be modeled using \( e \). In solving exponential equations, the natural logarithm helps in manipulating expressions with variables in the exponent.
We use \( \ln \) since it's the inverse operation of the exponential function \( e^{x} \). So if \( y = e^{x} \), then \( \ln(y) = x \). This property is crucial for extracting exponents in equations.

• Natural logarithms simplify solving equations with exponential functions.
• They are especially handy when dealing with growth or decay processes.

Solving Equations

To solve equations involving exponential expressions, we need to "undo" the exponential part. This is often done using logarithms. For instance, in the equation provided \( 100 = 25(1.5)^{t} \), our goal is to isolate \( t \).
First, we isolate the exponential term by dividing both sides by 25, giving us \( 4 = (1.5)^{t} \).
Next, we use the natural logarithm on both sides, leading to \( \ln(4) = \ln((1.5)^{t}) \). This step is crucial because it allows us to bring \( t \) down from the exponent.
Finally, by using properties of logarithms, we simplify and solve for \( t \) to find our answer. This step-by-step process is typical in solving exponential equations.

Properties of Logarithms

Logarithms have several properties that are incredibly useful when solving equations. One such property is the power rule, which states that \( \ln(a^b) = b \cdot \ln(a) \). This allows us to deal with exponents effectively by converting multiplication into addition, or division into subtraction.
In our example, we have \( \ln((1.5)^{t}) \), which we simplify using the power rule: \( \ln(4) = t \cdot \ln(1.5) \).
This property helps to simplify the equation so we can solve for \( t \) directly.

• Logarithms turn multiplication inside the log into addition, which is often easier to work with.
• They convert division into subtraction, simplifying complex expressions.

Knowing these properties is key to solving complex logarithmic and exponential equations efficiently.