Problem 110 - A Graphical Approach to College Algebra

Question

(a) Prove the quotient rule of logarithms. $$\log _{a} \frac{x}{y}=\log _{a} x-\log _{a} y$$ (b) Prove the power rule of logarithms. $$\log _{a} x^{r}=r \log _{a} x$$

Step-by-Step Solution

Answer

Quotient rule: \( \log_a \frac{x}{y} = \log_a x - \log_a y \). Power rule: \( \log_a x^r = r \log_a x \).

Step 1: Understand Logarithm Basics

Recall that by definition, \( \log_a x \) is the exponent to which the base \( a \) must be raised to obtain the number \( x \). Therefore, if \( \log_a x = m \), it means \( a^m = x \).

Step 2: Prove the Quotient Rule

Suppose \( \log_a x = m \) and \( \log_a y = n \). Then, by definition, \( a^m = x \) and \( a^n = y \). For the quotient \( \frac{x}{y} \), we have \( \frac{x}{y} = \frac{a^m}{a^n} \). This is equal to \( a^{m-n} \) by the laws of exponents. Thus, \( \log_a \frac{x}{y} = m - n = \log_a x - \log_a y \), proving the quotient rule.

Step 3: Convert Power Expression to Exponential Form

To prove the power rule, start with \( \log_a x^r = m \), which means \( a^m = x^r \). Using the properties of exponents, this is \( (a^m)^r = x^r \), which simplifies to \( a^{rm} = x^r \).

Step 4: Derive Power Rule from the Exponential Form

Thus, we have \( a^{rm} = (a^m)^r = x^r \). Therefore, \( rm = \log_a x^r \), and by the laws of logarithms, this gives \( \log_a x^r = r \log_a x \), confirming the power rule.

Key Concepts

Quotient Rule of Logarithms

The Quotient Rule of Logarithms helps simplify division inside a logarithm. This rule states that the logarithm of a quotient is equal to the difference of the logarithms. Specifically, for a given base \(a\), if you have \(\log_a \frac{x}{y}\), it can be expressed as \(\log_a x - \log_a y\).

To understand this concept better, consider the definitions of logarithms: \(\log_a x = m\) and \(\log_a y = n\). This means \(a^m = x\) and \(a^n = y\). If you were to divide \(x\) by \(y\), you have \(\frac{x}{y} = \frac{a^m}{a^n}\). This simplifies to \(a^{m-n}\) using properties of exponents, which can also be rewritten in logarithmic form as \(m - n\). Hence, \(\log_a \frac{x}{y} = \log_a x - \log_a y\). This process shows how the seemingly complex operation of dividing two numbers can become a straightforward subtraction task when using logarithms.

Power Rule of Logarithms

The Power Rule of Logarithms provides a way to handle exponents within logarithms. The rule states that the logarithm of a number raised to a power can be simplified to the power times the logarithm of the base number: \(\log_a x^r = r \log_a x\).

To see why this is true, start from the expression \(\log_a x^r = m\). This implies that \(a^m = x^r\) according to the definition of logarithms. By applying the rules of exponents, we have \((a^m)^r = x^r\), which simplifies to \(a^{rm} = x^r\). This equivalence means that the exponent \(rm\) is equal to the logarithm \(\log_a x^r\), giving us \(\log_a x^r = r \log_a x\).

In essence, the Power Rule takes potentially cumbersome calculations involving exponentiation and reduces them to a simple multiplication, making complex logs much more manageable. This is particularly useful in various mathematical and scientific applications where powers frequently appear.

Exponential Form in Logarithms

Exponential form is integral in proving logarithmic rules like the quotient and power rules. Understanding how to shift between exponential and logarithmic forms helps not only in algebraic manipulation but also secures a deeper grasp of how these mathematical conversions work.

When a logarithm, such as \(\log_a x\), equals a number, say \(m\), it tells us \(a^m = x\). This is the exponential form. This conversion is crucial because many properties of logarithms, including the quotient and power rules, depend on rewriting logarithmic expressions as exponents.

• Quotient Rule: When \(x \div y\) is expressed as \(\frac{a^m}{a^n}\), it simplifies to \(a^{m-n}\), demonstrating the rule \(\log_a \frac{x}{y} = \log_a x - \log_a y\).
• Power Rule: When \(x^r\) is expressed as \((a^m)^r\), it's simplified to \(a^{rm}\), leading to \(\log_a x^r = r \log_a x\).

Understanding these transformations helps pinpoint exactly why and how logarithmic rules work. By translating logarithmic expressions into exponential language, students can more easily track the logic that guides the rules of logarithms.