Problem 3 - Applied Calculus

Question

Solve for \(t\) using natural logarithms. $$2=(1.02)^{t}$$

Step-by-Step Solution

Answer

The solution is \(t \approx 35.003\).

Step 1: Set Up the Equation

We start with the given equation: \(2 = (1.02)^t\) Our goal is to solve for \(t\).

Step 2: Apply the Natural Logarithm

Take the natural logarithm of both sides of the equation to bring the exponent down. Applying \(\ln\):\(\ln(2) = \ln((1.02)^t)\).

Step 3: Use the Power Rule of Logarithms

Use the power rule \(\ln(a^b) = b\ln(a)\) to simplify the right side:\(\ln(2) = t \cdot \ln(1.02)\).

Step 4: Solve for t

To isolate \(t\), divide both sides by \(\ln(1.02)\):\(t = \frac{\ln(2)}{\ln(1.02)}\).

Step 5: Calculate the Value

Now calculate \(t\) using a calculator:\(t = \frac{\ln(2)}{\ln(1.02)} \approx 35.003\).

Key Concepts

Exponential Equations

Exponential equations are unique and powerful mathematical expressions where variables appear as exponents. In the given problem, we are presented with an equation of the form \(2 = (1.02)^t\). Here, the variable \(t\) is the exponent applied to the base 1.02. Exponential equations can model rapid growth or decay, such as population growth, radioactive decay, or interest compounding.

When solving exponential equations, a common technique is applying logarithms because they help in isolating the exponent. By using logarithms, we can convert an exponential equation into a more manageable linear form. This conversion simplifies the process of solving for the unknown exponent.

It's important to choose the correct logarithm. In this exercise, we use the natural logarithm (ln), a logarithm with base \(e\), where \(e\) approximates to 2.718. Natural logarithms are particularly useful in calculus and are often used with exponential growth or decay problems.

Power Rule of Logarithms

The power rule of logarithms is a manipulation tool aiding in simplifying expressions where exponents are involved. It states that \(\ln(a^b) = b \cdot \ln(a)\). This rule helps us pull down exponents, making them simpler to work with.

In our problem, we started with \(\ln((1.02)^t)\). By applying the power rule, we transformed this into \(t \cdot \ln(1.02)\). This step was crucial as it moves the variable \(t\) from an exponent to a coefficient, making it directly solvable through basic algebraic methods.

• Simplification: Reduces complex exponential problems into simpler, linear forms.
• Application: Widely used in solving and deriving exponential growth and decay equations.

This ability to simplify equations makes the power rule a cornerstone technique in logarithmic equations.

Solving Logarithmic Equations

Solving logarithmic equations involves strategic reasoning, primarily aimed at isolating the variable of interest. In logarithmic form, equations often require several steps of simplification and division to solve for the unknown.

The original problem requires finding \(t\) in \(\ln(2) = t \cdot \ln(1.02)\). To extract \(t\), we perform basic algebraic operations, specifically division. We divide both sides of the equation by \(\ln(1.02)\), effectively isolating \(t\) and providing an exact solution: \(t = \frac{\ln(2)}{\ln(1.02)}\).

Calculating further using a calculator provides an approximate value of \(t = 35.003\). This approach not only gives us the solution but illuminates how logarithms simplify and unravel exponential structures.

• Equation Rearrangement: Key step to isolate variables.
• Computation: Requires calculator for accuracy.

Recognizing these patterns and steps enriches your mathematical problem-solving toolkit.