Problem 27 - Contemporary Precalculus

Question

Prove the Quotient Law for Logarithms: For \(v, w>0\) \(\ln \left(\frac{v}{w}\right)=\ln v-\ln w .\) (Use properties of exponents and the fact that \(\left.v=e^{\ln v} \text { and } w=e^{\ln w} .\right)\)

Step-by-Step Solution

Answer

Question: Prove the Quotient Law for Logarithms using the given fact that \(v=e^{\ln v}\) and \(w=e^{\ln w}\). Answer: To prove the Quotient Law for Logarithms, which states that \(\ln\left(\frac{v}{w}\right)=\ln v-\ln w\), we used properties of exponents and logarithms to demonstrate that the logarithm of the quotient of two positive numbers is equal to the difference between the logarithms of those numbers.

Step 1: Write the Quotient Law for Logarithms

The Quotient Law for Logarithms states that for any positive numbers \(v\) and \(w\), the logarithm of their quotient is equal to the difference of their logarithms. In other words, $$ \ln \left(\frac{v}{w}\right)=\ln v-\ln w $$

Step 2: Use the properties of exponents

Recall that the logarithm is the inverse function of the exponent, which means that if \(y=\ln x\), then \(x=e^y\). Using this property, we have that $$ v=e^{\ln v} \quad \text{and} \quad w=e^{\ln w} $$

Step 3: Find a common exponent

Divide both sides of the first equation by \(w\), and use the second equation to rewrite \(w\) in terms of its logarithm: $$ \frac{v}{w} = \frac{e^{\ln v}}{e^{\ln w}} $$

Step 4: Use the properties of logarithms

When we divide exponential expressions with the same base, we can subtract the exponents. In this case, we subtract the logarithms of \(v\) and \(w\): $$ \frac{v}{w} = e^{\ln v - \ln w} $$

Step 5: Apply the natural logarithm to both sides

Now, apply the natural logarithm to both sides of the equation, using the fact that the logarithm is the inverse function of the exponent, meaning that they essentially cancel each other out: $$ \ln\left(\frac{v}{w}\right)=\ln\left(e^{\ln v-\ln w}\right) $$ Which simplifies to: $$ \ln\left(\frac{v}{w}\right)=\ln v-\ln w $$

Step 6: Conclusion

We have successfully proved the Quotient Law for Logarithms using properties of exponents and logarithms: $$ \ln\left(\frac{v}{w}\right)=\ln v-\ln w $$ This shows that the logarithm of the quotient of two positive numbers is equal to the difference between the logarithms of those numbers.

Key Concepts

Properties of Exponents

The properties of exponents are essential tools in mathematics, making complex calculations simpler. They include specific rules and properties that enable the manipulation and simplification of expressions with exponents:

• The Product Rule: When multiplying two exponents with the same base, you add the exponents. For instance, if you have base \(a\), then \(a^m \times a^n = a^{m+n}\).
• The Quotient Rule: This involves dividing two exponents with the same base, leading to the subtraction of their exponents: \(\frac{a^m}{a^n} = a^{m-n}\).
• The Power Rule: When raising an exponent to another exponent, you multiply the exponents: \((a^m)^n = a^{m\cdot n}\).
• The Zero Exponent Rule: Any base raised to the zero power equals one: \(a^0 = 1\).

Understanding these rules helps in the manipulation of logarithmic expressions. For example, recognizing that the expression \(e^{\ln v}\) means the exponential and logarithmic functions are inverses, simplifies proving the Quotient Law for Logarithms.

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a logarithm with the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. It serves as a crucial function in various areas of mathematics, particularly calculus.

• Natural logarithms are particularly useful when dealing with continuous growth processes or, for example, in the context of compound interest and population growth.
• The key feature of the natural logarithm is its inverse relationship with the exponential function: if \(y = \ln x\), then \(e^y = x\).

This inverse property allows for the simplification of equations where the exponents involve \(e\). For example, proving the Quotient Law for Logarithms involves setting \(v = e^{\ln v}\) and \(w = e^{\ln w}\), allowing us to handle complex operations more smoothly.

Logarithmic Identities

Logarithmic identities are instrumental in simplifying and transforming logarithmic expressions. These identities include several critical rules:

• Quotient Law: The main focus of this exercise, it shows that the log of a quotient is the difference of the logs: \(\ln\left(\frac{v}{w}\right) = \ln v - \ln w\).
• Product Law: This states that the logarithm of a product is the sum of the logarithms: \(\ln(v \times w) = \ln v + \ln w\).
• Power Law: It indicates that the logarithm of a power is the exponent times the logarithm: \(\ln(v^p) = p \cdot \ln v\).
• Change of Base Formula: This converts between bases: \(\log_b a = \frac{\log_k a}{\log_k b}\).

Mastering these identities is crucial for algebraic manipulations involving logarithms, as shown when proving the Quotient Law is simplified through understanding the inverse properties of exponential and logarithmic functions.

Inverse Function

Inverse functions effectively "undo" each other. In the realm of logarithms and exponents, this involves two primary functions:

• Logarithmic Function: The inverse of the exponential function, particularly when focused on the natural logarithm with base \(e\).
• Exponential Function: For a natural logarithm \(\ln x\), the corresponding exponential function is \(e^y = x\).

The relationship between these functions is pivotal in many mathematical applications. For instance, proving the Quotient Law for Logarithms relies on the fact that the natural logarithm is the inverse of the exponential function. This relationship enables transforming logarithmic expressions to their exponential form and vice versa, facilitating algebraic simplification. When we apply the natural logarithm to an expression like \(e^{\ln v - \ln w}\), the inverse relationship simplifies this clearly showing \(\ln v - \ln w\), illustrating how these concepts fit together in mathematics.