Problem 8 - 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. \(\log _{9}\left(\frac{9}{x}\right)\)

Step-by-Step Solution

Answer

The expanded logarithmic expression is: \(1 - \log_{9}(x)\)

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule states that \(\log _{b}(\frac{M}{N}) = \log _{b}M - \log _{b}N\). Use this rule to break down the given expression into: \(\log _{9}(9) - \log _{9}(x)\).

Step 2: Evaluate Logarithm of Base to the Base

The logarithm base \(b\) of any number to the base \(b\) is 1, because \(b^1 = b\). Thus, \(\log _{9}(9) = 1\). Substitute this result into the expression from Step 1, yielding: \(1 - \log_{9}(x)\).

Step 3: Finish

There are no further evaluation or expansions that can be made without more information on \(x\). With information about \(x\)'s value, one might potentially simplify further, but the final answer for our purposes is: \(1 - \log_{9}(x)\).

Key Concepts

Properties of Logarithms

Logarithms have a set of rules that make it easier to work with them, especially when you're expanding or simplifying expressions. Understanding these properties is key to addressing logarithmic problems effectively. Here are the most commonly used:

• Product Rule: This rule states that the logarithm of a product, \(\log_b(M \times N) = \log_b(M) + \log_b(N)\), allows us to split products into separate terms added together.


• Quotient Rule: This is the opposite of the product rule, used when you're dealing with division. The logarithm of a quotient is \(\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)\), allowing the division to be expressed as a difference.


• Power Rule: If you have a power, this rule helps break it down: \(\log_b(M^n) = n \cdot \log_b(M)\).

These properties simplify complex logarithmic expressions and are often the first step in any logarithm problem-solving approach.

Quotient Rule

The quotient rule is a fundamental property that is extremely useful for simplifying logarithmic expressions. It helps translate a division inside the log into a simple subtraction outside. In the exercise we examined,

• We began with the expression \(\log_{9}\left(\frac{9}{x}\right)\).


• By applying the quotient rule, we rewrote it as \(\log_{9}(9) - \log_{9}(x)\).

This transformation allows for easier handling of each part of the expression, specifically distinguishing between terms we can directly evaluate and those that require further simplification or dependence on unknowns such as \(x\). Understanding this rule is important because it regularly appears in logarithmic equations, making divided expressions more manageable.

Base Evaluation

Evaluating a logarithm's base is about simplifying the expression using properties inherent to logarithms. A classic result in log base evaluation is that the log of a base itself is 1. This is because any number raised to the power of 1 is itself, a crucial fact in our example.

• In the given task, \(\log_{9}(9) = 1\) because when you raise 9 to the power of 1, you get 9.


• This fact is universally true for any base, simplifying expressions where a logarithm equals its base.

By replacing \(\log_{9}(9)\) with 1, the expression becomes much simpler to handle, often allowing for a further reduction of the problem into its final form.

Logarithmic Simplification

Simplifying logarithmic expressions involves making them as concise as possible, often using known values and properties of logarithms. In our exercise, the last step yielded a final expression that was already as simplified as it could be without additional data about \(x\).

• We started with \(\log_{9}\left(\frac{9}{x}\right)\) and, through steps involving the quotient rule and base evaluation, reached the simplified form \(1 - \log_{9}(x)\).


• This transformation reduced the expression to the basics, substituting in known values like \(\log_{9}(9) = 1\).


• Further simplification depends on more information, such as the specific value of \(x\), which wasn't provided.

The skill of simplification involves recognizing when an expression has been reduced as far as it can go under the circumstances, highlighting the role of logarithmic properties in problem-solving.