Problem 50 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Use logarithms to solve the equation for \(t\). $$\frac{1}{3} e^{-3 t}=0.9$$

Step-by-Step Solution

Answer

The short answer to the given question is: \(t \approx -0.331\).

Step 1: Rewrite the equation using properties of logarithms

Rewrite the given equation using natural logarithm (ln): \Eq{ \frac{1}{3} e^{-3 t}=0.9 } First, let's isolate the exponential term by multiplying both sides by 3: \[ e^{-3 t}= 2.7 \] Now apply the natural logarithm to both sides: \[ \ln(e^{-3 t}) = \ln(2.7) \]

Step 2: Solve the equation for \(t\)

Recall the logarithmic property that states \(\ln(a^b) = b \ln(a)\). Use this property to rewrite the left side of the equation: \[ -3t\ln(e) = \ln(2.7) \] Since the natural logarithm of \(e\) is 1, \(\ln(e) = 1\), the equation simplifies to: \[ -3t = \ln(2.7) \] Finally, to find \(t\), divide both sides of the equation by \(-3\): \[ t = \frac{\ln(2.7)}{-3} \] Now, you can use a calculator to evaluate the expression and find the value of \(t\): \[ t \approx -0.331 \] So the solution for \(t\) is approximately \(-0.331\).

Key Concepts

Exponential Equations

Exponential equations are a type of mathematical expression where a variable is found in the exponent. They can describe scenarios like population growth, radioactive decay, and many other naturally occurring processes.

Here’s how you can solve an exponential equation:

• Start by simplifying the equation as much as possible. This often involves isolating the exponential term.
• Apply logarithms to both sides of the equation in order to bring down the exponent and make the equation solvable for the variable.
• Use properties of logarithms to simplify further, and then solve for the unknown variable.

In our example, we started with the equation \( \frac{1}{3} e^{-3t}=0.9 \). By multiplying both sides by 3, we isolated the exponential part to get \( e^{-3t} = 2.7 \). This step is crucial when dealing with exponential equations.

Understanding this process allows us to handle real-world problems where rates of change are exponential, making it a valuable skill in both academic and applied contexts.

Natural Logarithms

Natural logarithms (often abbreviated as ln) involve the number \( e \), an irrational constant approximately equal to 2.71828. The natural logarithm is essentially the inverse operation of the exponential function involving \( e \).

When dealing with exponential equations, natural logarithms are especially useful because they can simplify expressions involving \( e \). For instance, in the expression \( \ln(e^x) \), the \( \ln \) and \( e \) "cancel" out, leaving just \( x \) due to the property that \( \ln(e) = 1 \). This helpful property is precisely why we use natural logarithms in solving the given problem.

In our equation \( e^{-3t} = 2.7 \), applying the natural logarithm to both sides gives us \( \ln(e^{-3t}) = \ln(2.7) \). This step allows us to bring the negative exponent down as a multiplication factor, turning the equation into a manageable linear form. Recognizing when and how to use natural logarithms transforms complex exponentials into simpler linear equations that we can solve with basic algebra.

Properties of Logarithms

Logarithms have several properties that simplify the solution of mathematical problems, particularly those involving exponential terms. Key properties to remember include:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^p) = p \cdot \log_b(M) \)

These rules allow you to manipulate logarithmic expressions through multiplication, division, and exponents. In solving the given equation, we specifically used the power rule. We applied the natural logarithm to both sides, resulting in \( -3t \cdot \ln(e) = \ln(2.7) \), and simplified the expression using \( \ln(e) = 1 \).

Remembering these properties streamlines the process of solving complex equations like the one in our problem. They are fundamental tools you will encounter frequently in both pure mathematics and practical applications.