Problem 9 - 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. $$\log \left(\frac{x}{100}\right)$$

Step-by-Step Solution

Answer

\(\log (x) - 2\)

Step 1: Identify the Property

We identify the property of logarithms that will be used. The expression inside the log is a fraction, showing a division operation. The apt property to apply in such a scenario is 'log(a/b) = log(a) - log(b)'.

Step 2: Apply the Property

Let's apply the property to the given expression \( \log \left(\frac{x}{100}\right) \). This expands to \( \log (x) - \log (100) \).

Step 3: Simplify the Expression

The logarithm base 10 of 100 is 2, as 10^2 = 100. Hence, we can simplify \( \log (100) \) as 2. So, the final expanded expression is \( \log (x) - 2 \).