Tutorial Week 9 - Solving Logarithmic Equations

Key Concepts:

• Two distinct positive real numbers, p and q.
• Aim: Show either p = q^3 or q = p^3.

Steps to Solve the Equation:

1. Given Equation:

   3log_p(9) - 2log_q(27) - 2log_p^-1(q) = 81

2. Rewrite Logarithms:

   • log(9) as $3^2$
   • log(27) as $3^3$
   • log(81) as $3^4$

3. Transform Equation:

   6log_p(3) - 6log_q(3) + 8log_p(q^-1) = 3

4. Convert to Consistent Base (log_3):

   $C$(log_3(p) - log_3(q)) + 8log_3(p^-1q) = 3

5. Simplify Base Conversion:

   • log_3(p) = (1/log_p(3))
   • log_3(q) = (1/log_q(3))
   • log_3(p^-1q) = log_p^-1(q)

6. Simplify Terms:

   1/log_3(p) + 4/log_3(q) - 4/log_3(p^-1q) = 0

7. Let: log_p(3) = m and log_q(3) = n

   Substitute:

   6(1/m - 1/n) - 4/n = 0

8. Algebraic Manipulation:

   • Simplify Expression:

     (n-m)/mn = 4

   • Square Both Sides

     (n-m)^2 = 4mn

   • Expand and Simplify

     n^2 - 2mn + m^2 = 4mn

   Reduce and Factorize

   • 3n^2 - 10mn + 3m^2 = 0
   • (m-3n)(3m-n) = 0

9. Solutions:

   • Either m = 3n or 3m = n

   Therefore:

   • log_p(3) = 3log_q(3)

10.Solutions:

• Either p = q^3
• Or q = p^3

Conclusion:

• The equation simplifies to show that p equals q^3 or q equals p^3.
• Logarithmic properties and algebraic manipulation are key to solving.