Problem 9 - Algebra and Trigonometry

Question

\(7-18\) Evaluate the expression. $$ \log 4+\log 25 $$

Step-by-Step Solution

Answer

The expression evaluates to 2.

Step 1: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log a + \log b = \log (a \times b) \). In this case, we can use this rule on the given expression: \( \log 4 + \log 25 \). So, we apply the rule: \( \log 4 + \log 25 = \log (4 \times 25) \).

Step 2: Simplify the Product Inside the Logarithm

Next, calculate the product inside the logarithm: \( 4 \times 25 \). This equals \( 100 \). Now, the expression becomes \( \log 100 \).

Step 3: Evaluate the Logarithm

The base of the logarithm is 10 when not specified (common logarithm). We know that \( 10^2 = 100 \), hence \( \log_{10} 100 = 2 \). Thus, \( \log 100 = 2 \).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms simplifies the addition of two logarithms by combining the operands into a single logarithm of a product. This important rule states that \( \log a + \log b = \log (a \times b) \). Here's why it works: when you're adding two logarithms, you're essentially multiplying the numbers they refer to. For example, if you have \( \log 4 + \log 25 \), you can rewrite this as \( \log (4 \times 25) \).

• By applying this rule, you facilitate the solving process by reducing multiple logarithmic terms into one.
• This makes calculation easier, especially in algebra where combining terms is necessary.

Mastering this rule is vital because it simplifies seemingly complex logarithmic expressions, making them easy to evaluate.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves simplifying complex logarithms into understandable terms. Once you apply any necessary logarithmic rules, the next step is to evaluate or calculate the expression's value.
In our example, after applying the product rule: \( \log 4 + \log 25 = \log 100 \). Here, evaluating means calculating \( \log 100 \).

• Calculate the product or division within the logarithm if needed, as it simplifies the expression further.
• Recognize common outcomes like powers of 10 which relate directly to the common logarithm.
• Use basic exponential knowledge: if \( 10^2 = 100 \), then \( \log_{10} 100 = 2 \).

Practicing with simple numbers helps build intuition for more complex computations.

Common Logarithm

The common logarithm is a logarithm with base 10, usually written simply as \( \log \) when the base isn't specified. This is the most used logarithm in everyday contexts and is essential in scientific and engineering calculations.
When evaluating \( \log 100 \), we recognize that the base is 10. Given \( 10^2 = 100 \), this expression equals 2.

• The common logarithm simplifies many operations, especially for calculations involving powers of 10.
• Because of its prevalence, always assume a logarithm is common unless specified otherwise.
• Understanding log basics, like the common logarithm, bridges arithmetic and exponential equations effectively.

In essence, knowing the base and the relationship to powers of 10 lays the foundation for comprehending more elaborate logarithmic expressions.