Problem 130 - Algebra and Trigonometry

Question

If \(\log 3=A\) and \(\log 7=B,\) find \(\log _{7} 9\) in terms of \(A\) and \(B\).

Step-by-Step Solution

Answer

The final answer is \(2 \frac{A}{B}\)

Step 1: Express 9 in terms of 3

Since \(9 = 3^2\), we can substitute this into the expression to get \(\log_7 (3^2)\) which simplifies to \(2 \log_7 3\)

Step 2: Change the base

Using the change of base formula, we can write this as \(2 \frac{\log 3}{\log 7}\). Thus, our entire equation becomes \(2 \frac{\log 3}{\log 7}\) which is equivalent to \(2 \frac{A}{B}\). Here, the change of base formula \(\log_a b = \frac{\log_c b}{\log_c a}\) is used where c can be any base.

Step 3: Substitute back A and B

Using the original definitions of \(A\) and \(B\) as \(\log 3\) and \(\log 7\) respectively, substitute them back into the equation. This completes the conversion and yields the final answer: \(2 \frac{A}{B}\).