Problem 70 - College Algebra

Question

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \log x+\log 15+\log \left(x^{2}-4\right)-\log (x+2) $$

Step-by-Step Solution

Answer

The given logarithmic expression simplifies to \( \log{15x^{2} - 30x} \)

Step 1: Apply the Sum property of Logarithms

The sum of two logarithms with the same base can be written as a single logarithm with the same base and the argument as the product of the original arguments. Using this property, we get: \( \log{x} + \log{15} + \log{(x^{2} - 4)} = \log{(15x(x^{2} - 4))} \)

Step 2: Apply the Difference property of Logarithms

The difference of the logarithm with the same base can be written as a single logarithm with the same base and the argument as the quotient of the original arguments. Using this property, we get: \( \log{(15x(x^{2} - 4))} - \log{(x + 2)} = \log{\frac{(15x(x^{2} - 4))}{(x + 2)}} \)

Step 3: Simplify the Argument

Now simplify the expression in the logarithm. By simplifying, we obtain: \( \log{\frac{(15x(x + 2)(x - 2))}{(x + 2)}} = \log{(15x(x - 2))}\)

Step 4: Distribute the Variables

To write this expression in a simple format, distribute the variables. \( \log{(15x(x - 2))} = \log{(15x^{2} - 30x)} \)

Key Concepts

Properties of Logarithms

Logarithms have certain properties that are incredibly useful for simplifying and manipulating logarithmic expressions. One of the key properties is the **Product Property**. This property states that the logarithm of a product is equal to the sum of the logarithms of the factors:

• \( \log_b (MN) = \log_b M + \log_b N \)

This property is particularly helpful when you have multiple logarithmic terms that you're trying to simplify into a single logarithm. Another important property is the **Quotient Property**. This one tells us that the logarithm of a quotient is the difference between the logarithms:

• \( \log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N \)

Lastly, there is the **Power Property**, which comes into play when you are dealing with an exponent inside a logarithm:

• \( \log_b (M^p) = p \cdot \log_b M \)

Understanding these properties is essential, as they form the foundation for more complex logarithmic manipulation and are often used in mathematical proofs and problem-solving scenarios.

Condensing Logarithmic Expressions

Condensing logarithmic expressions allows us to transform an expression with multiple logarithmic terms into a single, compact logarithm. This process often involves using the properties of logarithms, such as the Product and Quotient Properties, to combine or simplify the terms. For example, if you have the expression \( \log x + \log 15 + \log (x^2 - 4) - \log (x + 2) \), the goal is to use these properties to combine these terms into a single logarithm.
By applying the Product Property, you can combine the sum of logarithms into the logarithm of a product:

• \( \log x + \log 15 + \log (x^2 - 4) = \log (15x(x^2 - 4)) \)

Next, you use the Quotient Property to handle the subtraction:

• \( \log (15x(x^2 - 4)) - \log (x + 2) = \log \left(\frac{15x(x^2 - 4)}{x + 2}\right) \)

Simplification of the resulting term may still be required, as seen in the original solution where it was reduced to \( \log (15x(x - 2)) \). Condensing logarithms in this way makes it easier to evaluate or use in further calculations.

Logarithmic Identities

Logarithmic identities are consistent rules that apply to all logarithms and simplify the process of solving log equations and transforming expressions. One fundamental identity is that the logarithm of one is always zero:

• \( \log_b 1 = 0 \)

No matter the base, this identity holds because any number raised to the power of zero equals one. Another useful identity is that the logarithm of the base itself is always one:

• \( \log_b b = 1 \)

These basic identities serve as key pieces when solving log equations, especially when evaluating expressions or converting them into a different form. Furthermore, understanding that changing the base of a logarithm can be managed through the **Change of Base Formula** is crucial:

• \( \log_b a = \frac{\log_k a}{\log_k b} \)

This formula is instrumental when you need to compute logarithms with bases that are not directly available on a calculator or during simplification steps. Logarithmic identities, when used alongside logarithmic properties, allow for greater flexibility and efficiency in mathematical analysis and problem-solving.