Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, condense to a single logarithm if possible. \(-\log _{b}\left(\frac{1}{7}\right)\)

Step-by-Step Solution

Answer

The expression condenses to \(\log_b(7)\).

Step 1: Apply Logarithm Property

Use the property of logarithms that states \(-\log_b(a) = \log_b\left(\frac{1}{a}\right)\) to simplify the negative sign in front of the log. The expression can be rewritten as \(\log_b(7)\) because \(\log_b\left(\frac{1}{7}\right) = -\log_b(7)\).

Step 2: Simplification

Now, since we applied the property correctly, the expression simplifies directly to \(\log_b(7)\). There are no further calculations needed, and we have condensed it into a single logarithm.

Key Concepts

Properties of Logarithms

Logarithms have a few key properties that make them extremely useful in simplification. These properties help transform complex expressions into simpler forms, often making calculations much easier. Let's look at some crucial properties:

• The Power Rule: This states that \(\log_b(a^c) = c \log_b(a)\). It allows us to bring an exponent down as a multiplier.
• The Product Rule: It states that the logarithm of a product is the sum of the logarithms, written as \(\log_b(x \, y) = \log_b(x) + \log_b(y)\).
• The Quotient Rule: This is the opposite of the product rule, stating \(\log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y)\).

The problem in the exercise uses another property: the property of negation, which is that \(-\log_b(a) = \log_b\left(\frac{1}{a}\right)\). By using this property, we transformed the original expression from having a negative sign to a positive logarithm of a reciprocal. This helped in simplifying the expression into \(\log_b(7)\), since \(\log_b\left(\frac{1}{7}\right) = -\log_b(7)\).

Change of Base Formula

The change of base formula is a powerful tool that allows us to compute logarithms with different bases using a calculator. Since calculators typically only provide logarithms in base 10 or base \(e\), this formula becomes handy.The general formula is:\[\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\]where \(c\) can be any base that is readily available, such as 10 or \(e\). This formula is especially useful if your calculations require a base that your calculator does not directly support.In the given exercise, although the base is \(b\), knowing how to switch bases might be useful in different scenarios, particularly when solving problems without a calculator. The change of base formula provides a way to represent logarithms in any scale, making various mathematical computations accessible.

Simplifying Expressions

Simplifying logarithmic expressions is often about making them look more manageable or combining them into a single term. This process frequently involves applying the properties of logarithms.In the original example provided, the expression \(-\log_b\left(\frac{1}{7}\right)\) is simplified using the negation property to \(\log_b(7)\), which is a single logarithm. Simplification often reduces complexity by combining terms or eliminating negative signs. Here are some general strategies for simplifying logarithmic expressions:

• Use properties of logarithms like product, quotient, and power rules to break down or combine logarithms.
• Convert complex fractions or roots into exponential form to apply the power rule.
• Search for any common terms that can simplify multiple parts of the expression when combined.
• Whenever possible, reduce expressions into a single logarithm to make further calculations easier.

The key to simplification is recognizing patterns or properties that transform the expression into a more usable or solvable form. By practicing these rules, students can become more adept at handling various logarithmic problems effortlessly.