Problem 7 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 3 through 9 , simplify the expression given. (a) \(10^{\log 2-\log 3}\) (b) \(e^{2 \ln 5-\ln 2}\)

Step-by-Step Solution

Answer

(a) The simplified form of the expression \(10^{\log 2 - \log 3}\) is \(2/3\). (b) The simplified form of the expression \(e^{2 \ln 5 - \ln 2}\) is \(25/2\).

Step 1: Simplify the expression (a)

By applying the logarithmic property \(\log_b mn = \log_b m - \log_b n\), the expression \(10^{\log 2 - \log 3}\) can be rewritten as \(10^{\log (2/3)}\). Using the property \(b^{\log_b x} = x\), we can simplify the expression to \(2/3\).

Step 2: Simplify the expression (b)

The given expression is \(e^{2 \ln 5 - \ln 2}\). The logarithmic property \(\log_b mn = \log_b m - \log_b n\) can be used to rewrite the exponent as \(\ln{25} - \ln{2}\), which simplifies to \(\ln{25/2}\). Following that, using the law \(e^{\ln x} = x\), this expression can be simplified to \(25/2\).

Key Concepts

Logarithmic Properties

Understanding logarithmic properties is crucial when simplifying complex logarithmic expressions. Logarithms, the inverse operations of exponentiation, convert multiplication into addition, division into subtraction, and exponential operations into multiplicative ones.

Two key logarithmic properties illustrated in the exercise are:

• The Quotient Rule: For any positive numbers \( m \) and \( n \) and any positive base \( b \), except when \( b = 1 \), we have \( \log_b m - \log_b n = \log_b (\frac{m}{n}) \).
• The Power Rule: This rule states that \( b^{\log_b x} = x \) for any positive number \( x \) and base \( b \), where \( b \) is not equal to 1.

By utilizing these properties, simplifying expressions becomes a systematic approach rather than guesswork. For example, in the exercise, to simplify \( 10^{\log 2 - \log 3} \), we apply the Quotient Rule to obtain \( 10^{\log (\frac{2}{3})} \), which simplifies to \( \frac{2}{3} \) using the Power Rule.

Exponents and Logarithms

The interplay between exponents and logarithms is a foundational concept in algebra. Exponents represent repeated multiplication, while logarithms answer the question: to what exponent must we raise a base to get a certain number? For the expression \( e^{2 \ln 5 - \ln 2} \), we engage this relationship.

Here's the step-by-step thought process:

• First, recognize that \( 2 \ln 5 \) can be rephrased as \( \ln 5^2 \) since taking the natural logarithm (ln) is the inverse process of exponentiation with base \( e \).
• Then, apply the logarithmic Quotient Rule to combine the terms into a single logarithm: \( \ln 5^2 - \ln 2 = \ln(\frac{25}{2}) \).
• Lastly, use the Power Rule by recognizing that \( e \) raised to the power of a natural logarithm of a number (\( e^{\ln x} \)) simplifies to the number itself, \( x \). So the given expression simplifies to \( \frac{25}{2} \).

This methodical approach turns what seems like a complicated equation into a simpler one that leverages the definitions of exponents and logarithms.

Natural Logarithm

The natural logarithm, represented by 'ln', is a logarithm with base \( e \), where \( e \) is Euler’s number, approximately 2.71828. It is of special importance because of its unique properties in calculus, especially dealing with growth processes and the area under hyperbolas.

In the context of the given exercise, understanding that \( e^{\ln x} = x \) is paramount. When we simplify \( e^{2 \ln 5 - \ln 2} \), we use this principle to bring the values out of the logarithmic function, reducing the expression to \( 25/2 \).

The natural logarithm has other properties, including the Product Rule (\( \ln(m \times n) = \ln(m) + \ln(n) \)) and the Power Rule (\( \ln(m^n) = n \ln(m) \)), which are useful in various mathematical and real-world applications. By mastering the natural logarithm, one gains a versatile tool for tackling a broad spectrum of problems in mathematics.