Problem 42 - 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 \(I .\) Where possible, evaluate logarithmic expressions. $$ \log 250+\log 4 $$

Step-by-Step Solution

Answer

The condensed form of \(\log 250 + \log 4\) is \(\log 1000\) and its evaluated form is 3.

Step 1: Identify the logarithmic property

The property to be used here is the product rule of logarithms, which states that the sum of two logarithms with the same base is equal to the logarithm of the product of the two numbers. Mathematically, it is expressed as \(\log_bM + \log_bN = \log_b(MN)\). This rule will be applied to the given expression \(\log 250 + \log 4\).

Step 2: Apply the logarithmic property

According to the property identified in the previous step, \(\log 250 + \log 4\) is equivalent to a single logarithm of the product of 250 and 4, which is \(\log (250*4)\).

Step 3: Evaluate the logarithmic expression

The single logarithm, \(\log (250*4)\), can be simplified into \(\log 1000\). Now, this logarithm can directly be evaluated, given that the base of the logarithm is 10 (since no base is expressed, it's assumed to be 10), and 10 to the power of 3 is 1000, so \(\log 1000 = 3\).

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a fundamental tool when working with logarithmic expressions. This property makes it possible to simplify expressions by transforming the sum of logarithms into a single logarithm of a product. If you have two logarithms with the same base, like \( \log_bM \) and \( \log_bN \), you can use this rule to condense them into one: \[ \log_bM + \log_bN = \log_b(M \cdot N) \].

For example, let’s take the expression \( \log 250 + \log 4 \). Using the product rule, this becomes \( \log(250 \cdot 4) \). This simplifies the expression significantly. It's a handy way to manage complex logarithmic calculations by reducing the number of terms.

In applying this rule, always ensure all terms share the same base. If they don’t, the product rule of logarithms can’t be used directly.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves transforming a longer logarithmic expression into a single, more manageable one. The key to condensing is using properties like the product, quotient, and power rules of logarithms. In our example, we primarily use the product rule.

Given \( \log 250 + \log 4 \), we can condense this into \( \log(250 \cdot 4) \). This results in \( \log 1000 \). By condensing, we reduce the complexity of the expression, which can simplify further evaluation or integration into larger calculations.

Always look for opportunities to apply these rules, as simplifying expressions is crucial in solving mathematical problems efficiently. Remember, condensing is about finding simpler forms and higher productivity in calculations.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions entails solving to find the numerical value of the expression. Once you’ve condensed an expression by employing the product rule, the next step is to evaluate it.

For \( \log 1000 \), note that by its condensed form, and assuming base 10, it simplifies easily. Consider that 10 raised to the power of 3 equals 1000, meaning \( \log_{10} 1000 = 3 \). This simplification is only possible because of the power of condensing and leveraging base knowledge.

• Always ensure the base of the logarithm is consistent for evaluation; if there's no base specified, it is typically assumed to be 10.
• The clearer your condensed expression, the simpler the evaluation process becomes.

Keep practicing these steps to become proficient at evaluating various logarithmic expressions, laying a solid foundation for tackling more complex problems.