Problem 7 - College Algebra Essentials

Question

Use properties of logarithms to expand logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. \(\log _{7}\left(\frac{7}{x}\right)\)

Step-by-Step Solution

Answer

The simplification of the given logarithmic expression \( \log_7(\frac{7}{x}) \) is \( 1 - \log_7(x) \).

Step 1: Apply the Quotient Rule

Use the quotient rule to expand the logarithmic expression. According to this rule, \( \log_7(\frac{7}{x}) \) is equivalent to \( \log_7(7) - \log_7(x) \).

Step 2: Simplify using Logarithm Basics

Remember that any logarithm with the same base and argument gives 1, i.e., \( \log_b(b) = 1 \). Thus, \( \log_7(7) = 1 \). Substitute this into the expression and rewrite it as \( 1 - \log_7(x) \).

Step 3: Final Solution

After all simplifications, the fully expanded and simplified form of the given logarithmic expression is \( 1 - \log_7(x) \).

Key Concepts

Properties of Logarithms

Logarithms have several properties that make them very useful in simplifying expressions and solving equations. Understanding these properties can significantly enhance your mathematical skill set. The properties of logarithms are:

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of the factors. It can be represented as \( \log_b(MN) = \log_b(M) + \log_b(N) \).
• Quotient Rule: This is the focus of our exercise. It indicates that the logarithm of a quotient is the difference between the logarithms of the numerator and denominator. It is represented as \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \).
• Power Rule: This rule tells us that the logarithm of an exponentiated number can be brought down as a multiplier. It is expressed as \( \log_b(M^n) = n \log_b M \).
• Identity Rule: This states that \( \log_b(b) = 1 \) because the base \( b \) raised to the power 1 equals itself, \( b \).
• Zero Rule: According to this rule, \( \log_b(1) = 0 \) because any base raised to the power of 0 gives 1.

These properties are essential for transforming and simplifying complex logarithmic expressions.

Quotient Rule

The quotient rule is a powerful property of logarithms that simplifies the log of a division into a subtraction. For a fraction inside a logarithm, the quotient rule allows you to separate the terms for easier manipulation. Using the quotient rule transforms \( \log_b \left(\frac{M}{N}\right) \) into two simpler logs: \( \log_b M - \log_b N \).
This property helps in breaking down expressions where division is involved, making them straightforward to handle.
In our example, we used the quotient rule on \( \log_7\left(\frac{7}{x}\right) \) to get \( \log_7(7) - \log_7(x) \).
This separation is crucial in simplifying the expression further, as it dissects the log into parts that can be individually addressed. The simplicity provided by the quotient rule is highly beneficial, especially for expressions that appear daunting at first glance.

Logarithm Basics

Logarithms might seem intimidating at first, but their basic properties are quite simple. At its core, a logarithm answers the question: "To what power must we raise the base to obtain a given number?"
For example, \( \log_7(7) = 1 \) is straightforward because 7 raised to the power of 1 equals 7.

• Base-Identical Property: If the base and the argument of the log are identical, such as \( \log_b(b) \), the result is always 1.
• This property arises directly from the definition of logarithms: \( b^1 = b \).

In our exercise, after applying the quotient rule to the expression \( \log_7\left(\frac{7}{x}\right) \), the term \( \log_7(7) \) simplifies to 1, showing how straightforward handling of logarithmic basics leads to simplification.
Understanding basic logarithm rules is essential as it allows you to evaluate parts of expressions mentally and verify solutions quickly, without the need for extensive calculations or a calculator.