Problem 3 - 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. $$ \log _{7}(7 x) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(\log_{7}(x) + 1\).

Step 1: Apply Logarithmic Product Rule

Firstly, apply the logarithmic product rule which indicates that \(\log_{b}(mn) = \log_{b}(m) + \log_{b}(n)\). In this case our \(m\) is \(7\), and \(n\) is \(x\), hence the expression is equivalent to \(\log_{7}(7) + \log_{7}(x)\).

Step 2: Simplify \(\log_{7}(7)\)

Simplify the first of the terms by using the property that \(\log_{b}(b) = 1\) for any positive \(b \neq 1\). So, \(\log_{7}(7)\) simplifies to \(1\). This leaves the result as \(1 + \log_{7}(x)\).

Step 3: Rearrange the Terms

For a clean representation and more standard appearance, the terms can be rearranged as \(\log_{7}(x) + 1\). It's the final expanded form of the logarithmic expression given in the exercise.