Problem 5 - Introductory and Intermediate Algebra for College Students 4th

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\log (1000 x)$$

Step-by-Step Solution

Answer

The expanded and evaluated logarithmic expression is 3 + \( \log(x) \)

Step 1: Recognize the properties of logarithms

The properties of logarithms relevant for this task include the quotient rule: \( \log_b{AB} = \log_b{A} + \log_b{B} \) and the fact that \( \log_b{A^n} = n \cdot \log_b{A} \).

Step 2: Apply the product rule

Given \( \log (1000x) \), we can separate 1000 and x as two arguments of a product. Thus, we can use the product rule and write: \( \log(1000x) = \log(1000) + \log(x) \)

Step 3: Evaluate \( \log(1000) \)

\(\log(1000)\) is in base 10, and since 1000 is \(10^3\), its logarithm will be 3. We then get \( \log(1000) + \log(x) = 3 + \log(x) \)