Problem 78 - Precalculus : Building Concepts and Connections

Question

The cumulative box office revenue from the movie Terminator 3 can be modeled by the logarithmic function $$R(x)=26.203 \ln x+90.798$$ where \(x\) is the number of weeks since the movie opened and \(R(x)\) is given in millions of dollars. How many weeks after the opening of the movie did the cumulative revenue reach \(\$ 140\) million? (Source: movies.yahoo.com)

Step-by-Step Solution

Answer

The cumulative revenue reached $140 million approximately in the \(e^{1.87699} \approx 6.53\) weeks since the opening of the movie.

Step 1: Setting Up the Equation

We first need to set up the equation, so we replace \(R(x)\) with $140 and solve for \(x\). The equation is now $140 = 26.203 \ln{x} + 90.798.

Step 2: Subtraction

Next, subtract 90.798 from both sides of the equation to get the logarithmic term by itself. This gives us \(49.202 = 26.203\ln{x}\).

Step 3: Division

Then, divide by 26.203 on both sides to isolate the term with \(\ln{x}\). This results in \(1.87699 = \ln{x}\).

Step 4: Exponentiation

Now we can get rid of the natural logarithm(ln) by exponentiating both sides with base \(e\) (Euler's number) since the inverse of the natural logarithm is the exponential function with base \(e\). This gives us \(e^{1.87699} = x\).

Key Concepts

Understanding Natural Logarithms

Natural logarithms are a type of logarithm with a special base, known as Euler's number, approximately equal to 2.71828, and it is often denoted as 'e'. Logarithms, including natural ones, are the inverses of exponentiation. In simple terms, while exponentiation allows you to see what number 'a' raised to the power 'n' produces, logarithms help you find 'n' when you know 'a' and the result of the exponential function.

When you see the notation \( \ln(x) \), it denotes a natural logarithm of \(x\). In practical terms, it answers the question: 'To what power must we raise 'e' to obtain \(x\)?' The natural logarithm of 'e' itself is 1 because \(e^1 = e\), and the natural logarithm of 1 is 0 because \(e^0 = 1\).

In the given exercise, the natural logarithm is used within the formula to model the cumulative box office revenue, signifying the relationship between the revenue and the time passed in a non-linear, logarithmic trend.

Exponentiation: The Inverse of Logarithms

Exponentiation is the mathematical operation that involves taking one number known as the base (in this context 'e') and raising it to the power of another number, known as the exponent. In the step-by-step solution, exponentiation is used as the key to solving the equation involving a natural logarithm. Specifically, exponentiation with base 'e' undoes the logarithmic operation.

For example, if \( \ln(x) = y \), then \( e^y = x \). This property is central to solving logarithmic equations because it allows us to switch from a logarithmic form to an exponential form, effectively letting us solve for \(x\). This process of converting a logarithm to an expression with an exponent is often called 'exponentiating', and it's what enables us to solve for the number of weeks it took for the Terminator 3 movie to reach a certain revenue milestone.

Modeling with Logarithms in Real Life

Modeling with logarithms is a powerful tool for representing real-world situations where growth or decay isn't constant but rather changes over time. Many phenomena in economics, biology, and even social sciences are best described using logarithmic models because they effectively capture the concept of diminishing returns or slow initial growth accelerating over time.

In the example of Terminator 3's cumulative box office revenue, the logarithmic model reflects how blockbuster movies tend to earn a lot quickly after release and then see their revenue increase at a slower pace as weeks pass. This kind of model allows analysts and forecasters to make predictions and understand patterns based on historical data. By solving such logarithmic equations, as in our exercise, we can answer practical questions about the future or past performance of such phenomena.