Problem 10 - 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. $$\log \left(\frac{x}{1000}\right)$$

Step-by-Step Solution

Answer

The expanded and evaluated expression is \(\log(x) - 3\).

Step 1 - Apply Logarithmic Properties

Begin by applying the logarithmic properties to expand the given logarithmic expression. According to the properties of log mentioned above, express the logarithmic fraction as a difference. Thus, \(\log \left(\frac{x}{1000}\right)\) can be written as \(\log(x) - \log(1000)\).

Step 2 - Evaluate Logarithmic Expression

After expanding the expression, it's now time to evaluate it without the use of a calculator. \(\log(1000)\) is an expression which can be evaluated manually since \(1000 = 10^3\). Thus, \(\log(1000) = 3\). Hence, our expanded and evaluated expression is \(\log(x) - 3\)