Problem 7 - Applied Calculus

Question

For Problems \(1-16,\) solve for \(t\) using natural logarithms. $$50=10 \cdot 3^{t}$$

Step-by-Step Solution

Answer

The solution for \(t\) is approximately 1.4649.

Step 1: Isolate the Exponential Expression

Start with the equation: \[50 = 10 \cdot 3^t\]To isolate the exponential expression, divide both sides by 10:\[\frac{50}{10} = 3^t\]This simplifies to:\[5 = 3^t\]

Step 2: Apply Natural Logarithm

Take the natural logarithm on both sides of the equation:\[\ln(5) = \ln(3^t)\]Using the logarithmic property \(\ln(a^b) = b\ln(a)\), we can rewrite the equation as:\[\ln(5) = t \cdot \ln(3)\]

Step 3: Solve for t

To find the value of \(t\), divide both sides by \(\ln(3)\):\[t = \frac{\ln(5)}{\ln(3)}\]Calculating using a calculator gives:\[t \approx \frac{1.6094}{1.0986} \approx 1.4649\]

Key Concepts

Exponential Equations

Exponential equations are central in many areas of mathematics and everyday applications, like population growth or radioactive decay. In an exponential equation, the unknown variable is found in the exponent, as shown in the exercise equation: \[50 = 10 \cdot 3^t\] Here, \(t\) is the exponent we need to solve for. A key strategy when dealing with exponential equations is to isolate the exponential part by performing operations that simplify it.

• Divide or multiply both sides to isolate the exponential expression.
• In this exercise, dividing by 10 simplifies things: \[5 = 3^t\]

Once isolated, you are ready to transform the expression using logarithms, a powerful tool in solving exponential equations.

Solving Logarithmic Equations

Solving logarithmic equations often involves applying logarithms to handle the exponent, as seen with the application of the natural logarithm in the exercise. By applying the natural logarithm (\(\ln\)), you capitalize on its properties to linearize the equation:\[\ln(5) = \ln(3^t)\]Using the property of logarithms, \(\ln(a^b) = b \ln(a)\), simplifies this to:\[\ln(5) = t \cdot \ln(3)\]

• The power of the expression becomes a coefficient through logarithmic properties, helping transform a multiplicative relationship into a linear one.
• In this specific case, natural logarithms are optimal because they pair well with exponential functions. Specifically, those involving base \(e\), but they are broadly useful as shown here with base 3.

After reaching a linear form, solving for the unknown resembles solving any typical algebraic equation.

Mathematics Problem-Solving

Mathematics problem-solving often requires a strategic movement through steps that build upon each part systematically. In this scenario, we addressed a problem using a methodical approach to find \(t\). First, isolate the variable, then transform the equation using effective tools like logarithms, and finally solve for the unknown.This exercise highlights key steps including:

• Ensuring the equation is in a suitable form to apply further mathematical operations.
• Recognizing which mathematical tool, like natural logarithms, best suits the problem.
• Applying algebraic techniques, i.e., simplifying using log properties, to achieve a solvable linear equation.

By following these techniques, solving complex equations becomes manageable, cultivating a deeper understanding and developing strong problem-solving skills.