Problem 41 - Precalculus

Question

Prove the Quotient Rule and Power Rule for Logarithms.

Step-by-Step Solution

Answer

Logarithm of a quotient is the difference, and logarithm of a power is the exponent multiplied.

Step 1: Recall the Properties of Exponents

To begin, let's recall the basic property of exponents which states that division of powers can be rewritten using subtraction: \( a^m / a^n = a^{m-n} \). This is fundamental in understanding the Quotient Rule for logarithms, as logarithms are essentially the inverse of exponentiation.

Step 2: Define the Quotient Rule for Logarithms

The Quotient Rule states that the logarithm of a quotient is the difference of the logarithms: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \). This means if you have two numbers, \(x\) and \(y\), and you want to find the logarithm of their division, you can subtract the logarithm of \(y\) from the logarithm of \(x\).

Step 3: Prove the Quotient Rule Using Exponents

Assume \( \log_b(x) = m \) and \( \log_b(y) = n \). This implies \( b^m = x \) and \( b^n = y \). The quotient \( \frac{x}{y} \) is \( \frac{b^m}{b^n} = b^{m-n} \). Logarithm on base \(b\) gives us \( \log_b\left(b^{m-n}\right) = m-n \). Thus, \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).

Step 4: Define the Power Rule for Logarithms

The Power Rule states that the logarithm of a power is the exponent times the logarithm: \( \log_b(x^c) = c \cdot \log_b(x) \). This means if you have a number \(x\) raised to a power \(c\), you can multiply \(c\) by the logarithm of \(x\).

Step 5: Prove the Power Rule Using Exponents

Assume \( \log_b(x) = m \), which means \( b^m = x \). Then, \( x^c = (b^m)^c = b^{mc} \). Taking the logarithm on both sides, \( \log_b(x^c) = \log_b(b^{mc}) = mc \) which simplifies to \( c \cdot \log_b(x) \). Therefore, \( \log_b(x^c) = c \cdot \log_b(x) \).

Key Concepts

Quotient Rule for Logarithms

The Quotient Rule for logarithms is a handy tool in mathematics that simplifies the process of finding the logarithm of a division. The rule states that the logarithm of a quotient is equivalent to the difference between the logarithms of the numerator and the denominator. In mathematical terms, this is expressed as:

• \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \)

To understand this rule better, consider two numbers, \(x\) and \(y\). When you want to determine the logarithm of \(x\) divided by \(y\), instead of performing division directly, you can simply subtract the logarithm of \(y\) from the logarithm of \(x\). This property stems from the inverse nature of logarithms to exponents. This means that just as multiplying numbers together relates directly to addition via logarithms, division correlates with subtraction.
For example, if you have \( \log_2(8) = 3 \) and \( \log_2(4) = 2 \), by applying the quotient rule, you find:

• \( \log_2\left(\frac{8}{4}\right) = \log_2(8) - \log_2(4) = 3 - 2 = 1 \)

This matches with \( \log_2(2) = 1 \), confirming the rule's accuracy.

Power Rule for Logarithms

The Power Rule for logarithms is an important principle allowing you to simplify logarithmic expressions involving exponents. According to this rule, if you have a number raised to a power, the logarithm of this power equals the exponent times the logarithm of the base number. The formula can be expressed as:

• \( \log_b(x^c) = c \cdot \log_b(x) \)

Let's break this down: suppose you have a number \(x\) and you want to find the logarithm of \(x\) raised to the power \(c\). Instead of computing \(x^c\) first and then finding the logarithm, the power rule allows you to simply multiply \(c\) by the logarithm of \(x\).
Consider the case where \( \log_2(3) = 1.585 \). If you want to find \( \log_2(3^2) \), applying the power rule gives:

• \( \log_2(3^2) = 2 \cdot \log_2(3) = 2 \cdot 1.585 = 3.17 \)

This rule highlights how logarithms simplify calculations involving powers by converting multiplicative relationships into additive ones.

Properties of Exponents

Understanding the properties of exponents is essential when working with logarithms, as they form the basis for both the quotient and power rules. One key exponent property is the division of like bases, which states:

• \( a^m / a^n = a^{m-n} \)

This indicates that when you divide exponents with the same base, you subtract their powers. This property is crucial because logarithms are the inverse functions of exponentiation, and these relationships provide the foundation for logarithmic identities.
To apply this in the context of logarithms, let's assume you know \( a^m = x \) and \( a^n = y \). If you wanted to express the division of these as a logarithm, the property becomes:

• \( x/y = a^{m-n} \)
• Taking the logarithm, you have \( \log_b\left(\frac{x}{y}\right) = m-n \)

This is precisely the foundation for the quotient rule for logarithms. Similarly, the property of multiplying exponents, \( (a^m)^c = a^{mc} \), is what simplifies the power rule, linking it to multiplication in the logarithmic world.