Problem 41 - Introductory and Intermediate Algebra for College Students 4th

Question

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. $$\log _{2} 96-\log _{2} 3$$

Step-by-Step Solution

Answer

The condensed expression equals \(5\).

Step 1: Apply the Quotient Rule

To begin, it's necessary to apply the quotient rule of logarithms. The quotient rule states that the difference of two logs with the same base equals the log of the quotient of their arguments. So, apply the quotient rule to the given expression \(\log_{2} 96 - \log_{2} 3\) to obtain \(\log_{2} (96/3)\).

Step 2: Evaluate the Inner Expression

Next, perform the division inside the logarithm from the previous step. So, 96 divided by 3 equals 32. Substituting this result back into the logarithm, the expression becomes: \(\log_{2} 32\).

Step 3: Evaluate Logarithmic Expression

In this stage, the goal is to find the exponent or power to which the base number (in this case 2) must be raised to obtain the number inside the logarithm (here, 32). By the definition of logarithms, this evaluates to 5 because 2 raised to the power of 5 equals 32. So, \(\log_{2} 32 = 5\).

Key Concepts

Properties of Logarithms

Logarithms are powerful tools in mathematics, often used to simplify complex expressions. They have unique properties which make manipulation possible and easier. One of the properties is that you can transform products, quotients, and powers into sums, differences, and products, respectively, of logarithms. These properties are essential when solving problems involving logarithmic expressions.

• Product Rule: This states that the logarithm of a product is the sum of the logarithms of the factors. Mathematically, it is expressed as: \[\log_b (MN) = \log_b M + \log_b N\]
• Quotient Rule: This allows you to express the logarithm of a quotient as the difference of the two individual logarithms, as we see in the problem: \[\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\]
• Power Rule: When a value is raised to a power, the logarithm can simplify to the power times the logarithm of the base number: \[\log_b (M^n) = n \cdot \log_b M\]

Understanding these properties helps in expanding or condensing logarithmic expressions, allowing for easier evaluation of complex logarithmic problems.

Quotient Rule

The quotient rule is an essential tool for simplifying logarithmic expressions that involve division. Essentially, it allows for the subtraction of two logarithms of the same base to be converted into a single logarithmic expression. This can be very useful in breaking down and solving logarithmic problems.
In our exercise, the quotient rule simplifies \(\log_{2} 96 - \log_{2} 3\) into \(\log_{2} \left(\frac{96}{3}\right)\).
This step converts the expression from a subtraction problem of two logarithms into a single logarithmic term.

• How it Works: Think about the division inside the logarithm as taking away a portion of the whole. The formula for the quotient rule is: \[\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N\]
• Why It's Useful: This rule makes it easier to further evaluate or simplify the expression, preparing it for the next steps.

The quotient rule, like other logarithmic properties, is very handy, especially when multiple logarithmic terms need to be condensed into one single expression.

Evaluate Logarithms

Evaluating logarithms is the process of finding the exponent that a base number is raised to in order to achieve the number inside the logarithm. This is an integral concept in logarithmic mathematics since it bridges the gap between logarithmic form and exponential form.
Using our solved example, we transformed \(\log_{2} \left(\frac{96}{3}\right)\) into \(\log_{2} 32\).
This means we are looking for a power, such that raising 2 to that power results in 32.

• Understanding Exponents: To solve this, you recognize that \(2^5 = 32\). Therefore, \(\log_{2} 32 = 5\). This means 2 must be raised to the 5th power to equal 32.
• Step by Step Approach: Always start with the known base and work through the possible powers until the number inside the logarithm is reached.

Evaluating logarithms connects the relationship between exponential and logarithmic forms, thereby enhancing understanding and solving of logarithmic equations.