Problem 35 - Elementary and Intermediate Algebra

Question

Write logarithm as a sum. Then simplify, if possible. \(\log 100 p q\)

Step-by-Step Solution

Answer

\( \log 100pq = 2 + \log p + \log q \)

Step 1: Apply the Product Rule

The product rule of logarithms states that \( \log(a \cdot b) = \log a + \log b \). Apply this to the expression \( \log(100pq) \), breaking it down into products: \( \log 100 + \log p + \log q \).

Step 2: Simplify \(\log 100\)

Recognizing \( 100 = 10^2 \), use the power rule of logarithms \( \log(a^b) = b \log a \) to simplify \( \log 100 \): \( \log 100 = \log 10^2 = 2\log 10 \).

Step 3: Substitute and Simplify the Expression

Substitute the simplification \( \log 100 = 2\) into the expression: \( \log 100 + \log p + \log q = 2 + \log p + \log q \).

Step 4: Conclusion: Simplified Expression

The expression \( \log 100pq \) as a sum of logarithms is \( 2 + \log p + \log q \).

Key Concepts

Understanding the Product Rule of Logarithms

The product rule of logarithms helps simplify expressions where logarithms involve products of numbers or variables. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors. Mathematically, it is expressed as:

• \( \log(a \cdot b) = \log a + \log b \)

For example, if you have \( \log(2 \cdot 5) \), using the product rule, you can break it down to \( \log 2 + \log 5 \). This makes it easier to work with and solve, especially if the individual logs are known or can be further simplified.
In the context of our exercise, applying the product rule to \( \log(100pq) \) involves breaking it into \( \log 100 + \log p + \log q \). This sets the stage for further simplification using other logarithm rules.

Decoding the Power Rule of Logarithms

The power rule of logarithms is another useful tool that simplifies expressions where a number or variable inside a logarithm is raised to an exponent. This rule indicates that if you have a logarithm of a number raised to a power, you can pull the exponent out as a multiplier in front of the log. The formula is:

• \( \log(a^b) = b \cdot \log a \)

For instance, \( \log(8^3) \) can be simplified to \( 3 \cdot \log 8 \) using the power rule. This makes calculations easier and often allows further simplification or evaluation.
In our exercise, recognizing that \( 100 = 10^2 \) allows us to use the power rule. Simplifying \( \log 100 \) to \( 2 \cdot \log 10 \), which is simply 2 if you know \( \log 10 \) equals 1, contributes to simplifying the expression further as part of a larger equation.

Crafting Logarithmic Expressions

Logarithmic expressions are mathematical phrases that involve logarithms and can be broken down, simplified, or evaluated using various logarithmic rules. Working with these expressions often requires:

• Applying the correct logarithmic rules; such as product, power, and quotient rules.
• Breaking down complex expressions into simpler parts.
• Understanding the properties of logarithms, such as \( \log 10 = 1 \).

In our problem, the expression \( \log 100pq \) initially appears complex, but using rules like the product and power rules helps transform it into a simpler form: \( 2 + \log p + \log q \).
Logarithmic expressions involve both constants and variables, and mastering manipulation of these expressions is key in algebra. Practice helps in retaining these laws and effectively applying them to solve problems in logarithmic form.