Solving Exponential Equations without Logs

Key Concepts

• Solving exponential equations often involves changing the base of the numbers involved in the equation so they are the same.
• When the bases are the same, the exponents can be set equal to each other.

Examples and Solutions

Example 1: Solving Without Logs

• Equation: (3^{x+2} = 9^{2x-3})
• Solution:
  • Change base: 9 is (3^2), so (9^{2x-3} = (3^2)^{2x-3} = 3^{4x-6}).
  • Set exponents equal: (x + 2 = 4x - 6).
  • Solve for (x):
    • Subtract (x) and add 6: (8 = 3x)
    • (x = \frac{8}{3})

Example 2: Using a Common Base

• Equation: (8^{4x-12} = 16^{5x-3})
• Solution:
  • Convert to base 2:
    • 8 is (2^3) and 16 is (2^4).
  • Rewrite equation: ((2^3)^{4x-12} = (2^4)^{5x-3})
  • Expand and equate exponents:
    • (12x - 36 = 20x - 12)
  • Solve for (x):
    • (-24 = 8x), (x = -3)

Example 3: Factorization Techniques

• Equation: (2^{x^2 + 3x} = 16)
• Solution:
  • Recognize 16 as (2^4).
  • Combine exponents: (x^2 + 3x = 4).
  • Factor quadratic: (x^2 + 3x - 4 = 0).
  • Solve: (x = -4) or (x = 1).

Example 4: Complex Fraction and Factoring

• Equation: (3^{2x} - 3^{2x-1} = 18)
• Solution:
  • Factor out (3^{2x}).
  • (3^{2x}(1 - \frac{1}{3}) = 18).
  • Simplify: (3^{2x} \times \frac{2}{3} = 27).
  • Solve: (3^{2x} = 27 = 3^3).
  • (2x = 3), (x = \frac{3}{2}).

Solving with Logs

• Example: (3^x = 8)
  • Take log of both sides and bring down exponent.
  • (x \log 3 = \log 8).
  • Solve: (x = \frac{\log 8}{\log 3}) ≈ 1.8928

Using Natural Logarithms for Base e

• Example: (e^x = 7)
  • Use natural log: (x \ln e = \ln 7).
  • (x = \ln 7) ≈ 1.9459

Conclusion

• Understanding how to manipulate bases and exponents is crucial for solving exponential equations without immediately resorting to logarithms.
• Factoring and substitution can be useful techniques in solving more complex equations.