Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.) $$\log _{6} a b^{3} c^{-2}$$.

Step-by-Step Solution

Answer

The expanded form of \( \log _{6} a b^{3} c^{-2} \) using the properties of logarithms is \( \log _{6} a + 3 \log _{6} b - 2 \log _{6} c \)

Step 1: Apply the multiplication property of logs

Substitute \( \log _{6} a b^{3} c^{-2} \) with \( \log _{6} a + \log _{6} b^{3} + \log _{6} c^{-2} \) using the multiplication property of logarithms.

Step 2: Apply the division property of logs

Redefine \( \log _{6} c^{-2} \) as \( -2 \log _{6} c \) using the power rule of logarithms.

Step 3: Apply the power rule of logs

Redefine \( \log _{6} b^{3} \) as \( 3 \log _{6} b \) using the power rule of logarithms.

Step 4: Final expression

Combine the expressions from step 2 and 3 to get the final expanded expression: \( \log _{6} a + 3 \log _{6} b - 2 \log _{6} c \)

Key Concepts

Properties of Logarithms

The properties of logarithms hold a key role in simplifying complex logarithmic expressions. They come in handy when you're faced with expressions involving products, quotients, or powers. The fundamental properties include:

• Product Property: This property states that the logarithm of a product is equal to the sum of the logarithms of the factors. Mathematically, it's expressed as: \( \log_b (MN) = \log_b M + \log_b N \).
• Quotient Property: According to this property, the logarithm of a quotient is the difference of the logarithms. It is denoted by: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \).
• Power Property: The power property lets you take the exponent in front of the log as a multiplier. It looks like: \( \log_b (M^p) = p \cdot \log_b M \).

Understanding these properties is crucial, as they allow us to break down complex expressions into simpler components. This makes it easier to handle and solve logarithmic problems.

Expanding Logarithmic Expressions

Expanding logarithmic expressions involves using the properties of logarithms to rewrite an expression as a series of simpler logarithms. Here's a look at how this works:
When you're given an expression, like \( \log_6 ab^3c^{-2} \), you can start by applying the Product Property. This means you express the logarithm of the entire product as a sum:
\[ \log_6 ab^3c^{-2} = \log_6 a + \log_6 b^3 + \log_6 c^{-2} \]
Next, use the Power Property to handle components with exponents. For example:

• Express \( \log_6 b^3 \) as \( 3 \log_6 b \).
• Express \( \log_6 c^{-2} \) as \( -2 \log_6 c \).

Once expanded, combining these expressions gives you the final expanded form:
\[ \log_6 a + 3 \log_6 b - 2 \log_6 c \]
This method of expansion proves useful in solving algebraic problems involving logarithms, providing a clearer path to simplify and manipulate expressions.

Logarithm Rules

Logarithm rules are essential tools for manipulating logarithmic expressions. Let's explore these necessary tools that help decode the complexities of logarithms:

• The Product Rule indicates that the log of a product is the sum of the logs. This means you can break a product inside a logarithm into separate terms.
• The Quotient Rule is about transforming a division inside a log into a subtraction: the log of the numerator minus the log of the denominator.
• The Power Rule suggests bringing down any exponent as a multiplier in front of the log term. This helps to eliminate powers within the log function.

All these rules are protocols that ensure consistency and correctness in mathematical operations involving logs. Applying these rules accurately yields results that are both simplified and manageable. For students, mastering these rules can open doors to a stronger grasp of mathematics, making complexities appear simple and approachable.