Problem 45 - 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 \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{2} 96-\log _{2} 3 $$

Step-by-Step Solution

Answer

The simplified version of the logarithmic expression \(\log _{2} 96-\log _{2} 3\) is 5.

Step 1: Apply the properties of logarithms

The laws of logarithms states that the difference of two logarithms on the same base can be written as the logarithm of the quotient. Using this property, the given expression \(\log _{2} 96-\log _{2} 3\) can be written as \(\log _{2} (96/3)\).

Step 2: Simplify the expression

Proceed to simplify the fraction inside the logarithm. \(96/3 = 32\). So, the expression now becomes \(\log _{2} 32\).

Step 3: Evaluate the logarithmic expression

Since the base of the logarithm is 2, \(\log _{2} 32\) asks for the exponent that would raise 2 to make 32. The number 32 can be obtained from 2 when you multiply 2 five times, therefore \(\log _{2} 32 = 5\)

Key Concepts

Properties of Logarithms

Learning the properties of logarithms can significantly simplify complex logarithmic expressions. The main properties include:

• Product Property: \( \log_b(MN) = \log_b M + \log_b N \)This property tells us that the logarithm of a product is equal to the sum of the logarithms of its factors.


• Quotient Property: \( \log_b(\frac{M}{N}) = \log_b M - \log_b N \)This indicates that the logarithm of a quotient equals the difference of the logs of the numerator and denominator.


• Power Property: \( \log_b(M^k) = k \cdot \log_b M \)This property shows that the logarithm of a number to a power is the exponent times the logarithm of the base number.

In the exercise, we used the Quotient Property to transform \( \log_2 96 - \log_2 3 \) into \( \log_2 (96/3) \). This drastically simplifies the expression, making it easier to handle or evaluate. Understanding and applying these properties is crucial in tackling logarithmic expressions efficiently.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves using logarithm properties to combine multiple logarithmic terms into a single expression. The goal is to create a singular, simplified log that can often be more straightforward to interpret or solve.
To condense a logarithmic expression, you typically follow these steps:

• Identify terms that can be combined using the properties of logarithms.
• Apply the Product, Quotient, or Power Property as needed.
• Simplify the resulting expression, ensuring all logs now have a coefficient of 1.

In our example, \( \log_2 96 - \log_2 3 \) was condensed by recognizing it as a subtraction that fits the Quotient Property, leading to\( \log_2 (96/3) \) which simplifies to \( \log_2 32 \). Breaking down the steps clearly and applying the right properties can make condensing a breeze.

Logarithmic Evaluation

Once a logarithmic expression is simplified, the next step is often to evaluate it. Evaluation means finding the numerical value that the logarithmic expression represents. When evaluating logarithms, you want to ask: "What power must the base be raised to in order to achieve this number?"
Consider \( \log_2 32 \).To evaluate it, determine what power the base 2 must be raised to equal 32.

• Start by considering 2 raised to different powers: 2¹ = 2,2² = 4,2³ = 8,2⁴ = 16,2⁵ = 32.


• Since 32 equals 2 raised to the power of 5, \( \log_2 32 \) equals 5.

This process forms the basis of logarithmic evaluation and helps to strengthen one's understanding of how logarithms work. Breaking down these calculations and thinking through the problem guides one to the correct answer and enhances mathematical intuition.