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

Step-by-Step Solution

Answer

The simplified form of \( \log _{7}\left(\frac{7}{x}\right) \) is \( 1 - \log_7(x) \).

Step 1: Apply the Quotient Rule

Rewrite the given expression \( \log _{7}\left(\frac{7}{x}\right) \) using the quotient rule of logarithms which states that \( \log_b(M/N) = \log_b(M) - \log_b(N) \). Thus, this expression becomes \( \log_7(7) - \log_7(x) \).

Step 2: Simplify Using the Property of Logarithms

Now, use the property \( \log_b(b) = 1 \), to simplify \( \log_7(7) \) to '1'. Hence, the expression further simplifies to \( 1 - \log_7(x) \).

Step 3: Final Simplified Expression

After simplification, the expanded expression of \( \log _{7}\left(\frac{7}{x}\right) \) is \( 1 - \log_7(x) \).