Problem 110 - A Graphical Approach to Precalculus with Limits

Question

When sunlight passes through lake water, its initial intensity \(I_{0}\) decreases to a weaker intensity \(I\) at a depth of \(x\) feet according to the formula $$\ln I-\ln I_{0}=-k x$$ where \(k\) is a positive constant. Solve this equation for \(I .\)

Step-by-Step Solution

Answer

The intensity \( I \) is given by the equation \( I = I_0 \cdot e^{-kx} \).

Step 1: Understand the Given Equation

We are given the equation \( \ln I - \ln I_0 = -kx \), which is a logarithmic equation describing the intensity of light as it passes through lake water. Our goal is to solve this equation for \( I \).

Step 2: Combine Logarithms Using Properties

Recognize that \( \ln I - \ln I_0 \) can be rewritten using the property of logarithms \( \ln \frac{a}{b} = \ln a - \ln b \). Thus, the equation becomes \( \ln \frac{I}{I_0} = -kx \).

Step 3: Exponentiate to Remove Logarithm

To eliminate the natural logarithm, exponentiate both sides of the equation. This gives \( \frac{I}{I_0} = e^{-kx} \).

Step 4: Solve for I

Multiply both sides of the equation by \( I_0 \) to isolate \( I \). This results in \( I = I_0 \cdot e^{-kx} \).

Key Concepts

Logarithmic Functions

Logarithmic functions are fundamental in understanding exponential growth and decay phenomena. They help us in handling equations where the unknown variable is an exponent. In its simplest form, a logarithm answers the question: "To what exponent must a base be raised to produce a given number?" In mathematical terms, if we have \( b^y = x \), then \( \log_b(x) = y \). Thus, a logarithmic function is essentially the inverse of an exponential function.

### Simplifying with Logarithmic PropertiesLogarithmic properties are very useful in simplifying expressions and solving equations. A key property is that the difference of two logarithms can be expressed as a quotient: \( \ln a - \ln b = \ln \frac{a}{b} \). This property was used in our exercise to rewrite the initial equation as \( \ln \frac{I}{I_0} = -kx \). Understanding these properties allows us to manipulate and solve complex logarithmic expressions easily.

Natural Logarithm

The natural logarithm, denoted by \( \ln \), is a logarithm with base \( e \), where \( e \approx 2.71828 \). It is widely used in science and engineering because of its natural properties related to growth processes. The natural logarithm simplifies modeling real-world phenomena, such as population growth, radioactive decay, and in this case, light intensity decay.

### Applying Natural Logarithms in CalculationsWhen you encounter equations with \( \ln \), and you need to solve for a variable in the exponent, taking the exponential of both sides cancels out the logarithm. In our exercise, by exponentiating both sides of \( \ln \frac{I}{I_0} = -kx \), we revert back to the exponential form: \( \frac{I}{I_0} = e^{-kx} \). This step is crucial as it transitions the equation from logarithmic terms to a solvable linear equation form where we can isolate and solve for \( I \).

Properties of Exponents

Properties of exponents are essential when transforming equations involving powers or repeating multiplication into simpler forms. Generally, if you have \( a^m \, \cdot \, a^n \), you can combine them as \( a^{m+n} \), or if you divide \( a^m / a^n \), it simplifies to \( a^{m-n} \). This toolkit allows for reduction and simplification of complex exponential expressions.

### Implementing Exponent ConceptsIn the context of our exercise, once we have transformed the equation into an exponential form \( \frac{I}{I_0} = e^{-kx} \), understanding that \( a^n \) can be rewritten for solving complex equations made it possible to clearly express \( I \) as \( I_0 \, e^{-kx} \). This representation is critical in calculating the light intensity at any depth \( x \), by substituting known values for \( I_0 \), \( k \), and \( x \). These exponential transformations make such calculations straightforward, providing insight into how changes in depth affect light intensity.