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

Step-by-Step Solution

Answer

The expansion of the given logarithmic expression \(\log \left(\frac{x}{100}\right)\) is \(\log(x) - 2\).

Step 1: Apply the Quotient Rule

To start with, the equation is written in logarithmic form as: \( \log \left(\frac{x}{100}\right) \)By applying the quotient rule, the equation becomes: \( \log(x) - \log(100) \)

Step 2: Evaluate Logarithmic Expressions

We know that \(\log(100) = 2\) because \((10^2 = 100)\), hence, replace \(\log(100)\) by \(2\) in the equation. So the equation now is: \( \log(x) - 2 \)