Problem 34 - College Algebra

Question

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm, and then round to three decimal places. $$ y=4.5(0.6)^{x} $$

Step-by-Step Solution

Answer

The simplified equation in terms of base \(e\) and expressed in terms of a natural logarithm is \(y = 4.5e^{x ln(0.6)}\).

Step 1: Rewrite the equation

Rewrite equation from base \(0.6\) to base \(e\), using the property \(b^x = e^{x ln b}\), which can be used to convert from one base to another by involving the natural logarithm. Thus, the equation \(y=4.5(0.6)^x\) can be rewritten as \(y = 4.5e^{x ln(0.6)}\).

Step 2: Simplify the equation

Simplify the equation using the properties of logarithms to remove the natural logarithm from the exponent. Apply the formula \(e^{a ln b} = e^{ln b^a} = b^a \). Thus, the equation simplifies to \(y = 4.5(0.6)^x\).

Step 3: Describe the final form

Now, the equation is in the right form, with base \(e\) and expressed in terms of a natural logarithm. However, the exercise asks for a value rounded to 3 decimal places. As there are no numbers in the equation that require rounding, no further steps are required.

Key Concepts

Understanding Base e

The mathematical constant known as base e, approximately equal to 2.71828, is fundamental to natural logarithms and exponential functions. It's a unique number with the property that the rate of exponential growth of the function e^x is equal to the function's current value. This means that if you were to graph this function, the slope at any point would match the function's value at that point.

When an exponent is raised to the base e, such as in e^x, it's typically dealing with growth processes, like population growth, compounded interest, or radioactive decay. In the given exercise, converting the original base 0.6 to base e, involves using the natural logarithm, which is a logarithm with base e, and is often denoted as ln(x). The transformation of the equation demonstrates how a function with a base other than e can be expressed in terms of the natural logarithm to simplify calculations and provide a clearer understanding of the function's growth rate.

Logarithm Properties

The properties of logarithms are immensely helpful in solving and simplifying equations. Logarithms, in essence, are the inverse of exponentiation, answering the question: To what power must a certain base be raised to obtain a given number? The properties include product, quotient, and power rules. For instance:

• The logarithm of a product is the sum of the logarithms (ln(ab) = ln(a) + ln(b)).
• The logarithm of a quotient is the difference between the logarithms (ln(a/b) = ln(a) - ln(b)).
• The power rule states that an exponent can be taken out in front of the logarithm (ln(a^b) = b · ln(a)).

In the exercise, these properties enable us to manipulate the equation so that we can express it using base e. The step that converts (0.6)^x to e^{x ln(0.6)} leverages the power rule, effectively switching the base from 0.6 to e, which is essential for further calculations and analyses in higher mathematics.

Exponential Functions

Exponential functions are characterized by their variable exponent, like y = b^x, where b is a positive real number. In real-world contexts, these functions model phenomena that grow or decay at rates proportional to their size - a principle seen in fields from finance to physics. The function's base determines how quickly it grows or decays. When the base is greater than 1, the function exhibits growth; when the base is between 0 and 1, as in the exercise, it represents decay.

The exercise provided uses the decay function 4.5(0.6)^x and by applying properties of logarithms we've expressed it in terms of e, the most natural base for growth and decay processes. This kind of transformation is practical because it allows using tools of calculus, such as differentiation and integration, which are more straightforward with base e. Simplifying an equation to its exponential form in base e gives a clearer view of the rate of change which is essential for solving many real-world problems.