Problem 31 - Elementary and Intermediate Algebra

Question

Write logarithm as a sum. Then simplify, if possible. \(\log _{2}(4 \cdot 5)\)

Step-by-Step Solution

Answer

\( \log_2(4 \cdot 5) = 2 + \log_2(5) \)

Step 1: Identify Logarithmic Property

The given problem is to express \( \log_2(4 \cdot 5) \) as a sum. The product inside the logarithm allows us to use the properties of logarithms, specifically the Product Rule: \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \).

Step 2: Apply the Product Rule

Using the Product Rule for logarithms, split \( \log_2(4 \cdot 5) \) into: \( \log_2(4) + \log_2(5) \). This expression separates the logarithm of a product into a sum of logarithms.

Step 3: Simplify the First Term

Recognize that \( 4 \) is a power of \( 2 \), specifically \( 2^2 \). Therefore, use the Power Rule of logarithms: \( \log_b(b^k) = k \cdot \log_b(b) \). Thus, \( \log_2(4) = \log_2(2^2) = 2 \cdot \log_2(2) = 2 \).

Step 4: Combine and Simplify If Possible

Substitute the simplified value of \( \log_2(4) \) back into the expression from Step 2 to get: \( 2 + \log_2(5) \). This is the simplified form as \( \log_2(5) \) does not simplify further using integer exponents.

Key Concepts

Product Rule for Logarithms

When faced with multiplying numbers inside a logarithm, a useful property comes to our aid: the product rule for logarithms. This rule allows us to express the logarithm of a product as the sum of the logarithms of the individual factors. For a base \( b \), this rule is stated as:

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

In simpler terms, if you have the logarithm of a multiplication, you can split it into the sum of two separate logarithms. In our original exercise, we used this rule to transform \( \log_2(4 \cdot 5) \) into \( \log_2(4) + \log_2(5) \). By this method, each part of the product is easier to analyze and further simplified.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions often involves using various properties of logarithms to reduce expressions to a more manageable form. This is crucial, as it makes complex expressions easier to evaluate or compare. During simplification, it's important to look out for opportunities to apply properties such as the product rule or power rule.

• Check if a number is a power of the base of the logarithm. It can be easily simplified using the power rule.
• Combine like terms if possible to reduce redundancy.
• Evaluate any known logarithmic values that yield integers to simplify calculations.

For example, in the original problem, once we split \( \log_2(4 \cdot 5) \) using the product rule, we noticed that 4 can be rewritten as \( 2^2 \). This observation led to further simplification using the power rule, greatly simplifying the expression.

Power Rule for Logarithms

The power rule for logarithms is another very handy property that helps in simplifying expressions. This rule lets you take the power, which is a part of the number you're taking the logarithm of, and bring it in front of the logarithm as a multiplier. The power rule can be illustrated as follows:

• \( \log_b(b^k) = k \cdot \log_b(b) \)

In essence, if a number inside your logarithm is an exponent of the logarithm's base, you move that exponent out front. In our example, \( 4 = 2^2 \), so with the base of \( 2 \), it simplifies easily using this rule: \( \log_2(4) = 2 \cdot \log_2(2) \), and since \( \log_2(2) = 1 \), this further simplifies to 2. This application of the power rule makes tackling logarithmic problems more straightforward and efficient.