Problem 10 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{1000}\right) $$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \( \log \left(\frac{x}{1000}\right) \) is \( \log(x) - 3 \).

Step 1: Apply the Quotient Rule

By the quotient rule of logarithms, \( \log \left(\frac{x}{1000}\right) \) can be written as \( \log(x) - \log(1000) \)

Step 2: Evaluate the Logarithm of 1000

We know that \( \log(1000) = \log(10^3) \). According to the rules of logarithms, this can be rewritten as \( 3 \log(10) \). The logarithm base 10 of 10 is 1, so \( 3 \log(10) = 3 \times 1 = 3 \). The expression now becomes \( \log(x) - 3 \)