Problem 7 - 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. $$ \log _{7}\left(\frac{7}{x}\right) $$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \( \log_7(7/x) \) is \(1 - \log_7(x)\).

Step 1: Identify log division rule

Let's start off with the logarithm division rule that states \( \log_b(a/c) = \log_b(a) - \log_b(c) \) where b is the base, and a and c are the arguments.

Step 2: Apply log division rule to simplify expression

Here, the give expression is \( \log_7(7/x) \). We can rewrite this as \( \log_7(7) - \log_7(x) \) according to our division rule.

Step 3: Evaluate the Logarithm

We know that logarithm of any number to the base of the same number equals 1. Hence, \( \log_7(7) = 1 \). Thus, the expression becomes \(1 - \log_7(x)\).