Problem 58 - Algebra 2

Question

Write \(\frac{\log _{5} 9}{\log _{5} 3}\) as a single logarithm.

Step-by-Step Solution

Answer

The expression is \( \log_{3} 9 \).

Step 1: Understand the Logarithmic Fraction

The given expression is \( \frac{\log_{5} 9}{\log_{5} 3} \). It represents a fraction where both numerator and denominator are logarithms with the same base (base 5).

Step 2: Apply the Logarithm Quotient Rule

According to the properties of logarithms, the expression \( \frac{\log_{b} a}{\log_{b} c} \) can be simplified to \( \log_{c} a \). In this case, \( b = 5 \), \( a = 9 \), and \( c = 3 \).

Step 3: Simplify Using the Rule

Substitute \( a = 9 \) and \( c = 3 \) into the rule: \( \frac{\log_{5} 9}{\log_{5} 3} = \log_{3} 9 \). Now the expression is written as a single logarithm with base 3.

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them an essential tool in mathematics, especially when dealing with equations involving exponents. Understanding these properties can help you manipulate and simplify logarithmic expressions effectively.

Here are some important properties of logarithms:

• Product Rule: The logarithm of a product is the sum of the logarithms. Symbolically, \( \log_{b}(mn) = \log_{b} m + \log_{b} n \).
• Quotient Rule: The logarithm of a quotient is the difference of the logarithms. This is particularly useful in simplifying expressions like the one in the exercise (more about this rule in the next section).
• Power Rule: The logarithm of a power is the exponent times the logarithm of the base. It can be represented as \( \log_{b}(m^n) = n \cdot \log_{b} m \).
• Change of Base Rule: A useful property to convert logarithms from one base to another by using a different base. We will discuss this more in the subsequent sections.

These properties provide the tools you need to solve various mathematical problems involving logarithms. By breaking down complex expressions into simpler ones, you make problems easier to manage and understand.

Logarithm Quotient Rule

The quotient rule of logarithms is one of the key tools for simplifying fractions of logarithms. This rule states that the logarithm of a division is the subtraction of the logarithms of the two numbers.

In formulaic terms, the quotient rule is presented as follows:

• \( \log_{b} \left( \frac{a}{c} \right) = \log_{b} a - \log_{b} c \)

Alternatively, when you have a fraction where both terms are logarithms with the same base, such as \( \frac{\log_{b} a}{\log_{b} c} \), it can be rewritten as a single logarithm: \( \log_{c} a \).

This rule is particularly handy in simplifying the expression given in the exercise:

• \( \frac{\log_{5} 9}{\log_{5} 3} = \log_{3} 9 \)

It transforms the original fraction into a single, more manageable logarithm with base 3, showing how powerful these rules can be in simplifying expressions.

Change of Base Formula

The change of base formula is a valuable tool for converting logarithms from one base to another. This is particularly useful because calculators typically only provide logarithms for the common base (base 10) and the natural base (base \(e\)).

The change of base formula is represented as:

• \( \log_{b} a = \frac{\log_{c} a}{\log_{c} b} \)

This formula allows you to take a logarithm with an uncommon base and rewrite it using a base that is more convenient, often simplifying calculations. In practice, you can choose base 10 or base \(e\) and use the calculator to evaluate these.

To put the change of base formula into context, assume you encounter a logarithm in base 5: \( \log_{5} 9 \). If you prefer to work in base 10:

• \( \log_{5} 9 = \frac{\log_{10} 9}{\log_{10} 5} \)

This method makes it easier to compute the logarithm using a calculator. Understanding the change of base formula and how to apply it will enhance your ability to work with logarithmic expressions effectively.