Problem 53 - College Algebra Essentials

Question

In Exercises \(53-56,\) rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$ y=100(4.6)^{x} $$

Step-by-Step Solution

Answer

The exponential equation \(y=100(4.6)^{x}\) when expressed with the base \(e\) and rounded to three decimal places is \(x = (\ln(y) - \ln(100)) / \ln(4.6)\).

Step 1: Identify known variables

From the equation \(y=100(4.6)^{x}\), it is identified that the base of the exponential equation here is \(4.6\), and the unknown here 'x' is the power to which the base is raised.

Step 2: Apply natural log on both sides

Take the natural logarithm on both sides of the equation to get: \(\ln(y) = \ln(100(4.6)^{x})\). This brings out the variable \(x\) from the exponent.

Step 3: Use properties of logarithms to simplify

Using the properties of logarithms, the equation can be simplified as: \(\ln(y) = \ln(100) + \ln((4.6)^{x})\). Applying power rule of logarithms, we can bring \(x\) in front of the log term resulting in: \(\ln(y) = \ln(100) + x \cdot \ln(4.6)\). Now, isolate \(x\) to get the equation in terms of \(x\): \(x = (\ln(y) - \ln(100)) / \ln(4.6)\).

Key Concepts

Understanding the Natural Logarithm

The natural logarithm, commonly represented as \(\ln\), is a special type of logarithm that has the mathematical constant \(e\) as its base. The number \(e\) is approximately equal to 2.71828 and is irrational, meaning it can't be represented as a simple fraction and its decimal goes on forever without repeating. The natural logarithm of a number \(x\) answers the question: 'To what power must \(e\) be raised, to produce the number \(x\)?' This concept is foundational when dealing with growth processes, compounding interest, or any phenomena following a continuous growth pattern.

For example, if you have an equation \(e^y = x\), taking the natural logarithm of both sides will reveal that \(y = \ln(x)\), essentially 'unpacking' the exponent on \(e\) and expressing it in its most direct form. The natural logarithm is a critical tool for solving exponential equations because it allows us to isolate the exponent when the base of the exponential expression is \(e\).

Leveraging Properties of Logarithms

Logarithms have several properties that are useful in simplifying and solving equations. Understanding these properties can greatly aid in working with complex expressions.

Firstly, the product rule: \(\ln(ab) = \ln(a) + \ln(b)\), which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Secondly, the quotient rule: \(\ln(\frac{a}{b}) = \ln(a) - \ln(b)\), implies that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. Lastly, one of the most powerful tools is the power rule: \(\ln(a^b) = b \cdot \ln(a)\), which allows us to bring down an exponent to the front of the logarithm, transforming an exponentiation into a multiplication. These properties are essential when simplifying logarithmic expressions and, as illustrated in the exercise's solution, when isolating variables in exponential equations.

Solving Exponential Equations with Logarithms

Exponential equations, where variables reside in the exponent, can often be challenging. However, by utilizing the natural logarithm and its properties, these can be transformed into more manageable forms. To solve an exponential equation, you want to isolate the variable of interest, generally through the use of logarithms.

Here are the steps to achieve this: First, apply the natural logarithm to both sides of the equation as this can help to 'extract' the variable from the exponent. Next, use the relevant properties of logarithms to simplify the equation—this might involve using the power rule to move exponents to the front or the product rule to separate multiplied terms. Lastly, isolate the variable by performing algebraic operations, such as adding, subtracting, dividing, or multiplying. As demonstrated in the solution for the exercise \(y=100(4.6)^x\), taking the natural logarithm of both sides followed by the application of logarithmic properties allowed for the isolation and solving for \(x\).