In-Depth Notes on Exponential Equations

Exponential Equations Learning Outcomes

  • Understand techniques for solving exponential equations involving common bases.
  • Identify situations when an exponential equation has no solution.
  • Employ logarithms to solve exponential equations.

One-to-One Property of Exponential Functions

  • Given real numbers ( b > 0 ) and ( b \neq 1 ):
    • If ( b^S = b^T ), then ( S = T ).
    • Applies to algebraic expressions as well.
Example: Solving Exponential Equations with Common Bases

  • Consider the equation:
    [ 3^{4x - 7} = 3^{2x} ]

    • Rewrite both sides using the common base (here, 3):
      [ 4x - 7 = 2x ]
    • Apply the one-to-one property by equating the exponents:
      [ 4x - 7 = 2x ]
    • Rearranging gives:
      [ 4x - 2x = 7 ]
    • This leads to:
      [ 2x = 7 \implies x = 3.5 ]

Recognizing No Solutions in Exponential Equations

  • Exponential functions only produce positive outputs. If the equation yields a negative solution, then it has no solutions.
Example:
  • Consider the equation: [ 3^{x + 1} = -2 ]
    • Real value of ( x ) cannot satisfy this, as the left-hand side is always positive.

Using Logarithms to Solve

  • When common bases are elusive, use logarithms:
    • Apply log to both sides:
      [ \log(b^S) = \log(b^T) \implies S \log b = T \log b ]
    • Utilize properties of logarithms to isolate variable terms.
Example: Solving with Logarithms
  • For ( 5^{x + 2} = 4^{x} ):
    1. Apply logarithm:
      [ (x + 2) \log 5 = x \log 4 ]
    2. Rearranging gives:
      [ x \log 5 + 2\log 5 = x \log 4 ]
    3. Grouping terms leads to:
      [ x(\log 5 - \log 4) = -2 \log 5 ]
    4. Finally, solve for ( x ).

Exponential Equations with e

  • For equations of the form: [ y = A e^{kt} ]
    • Steps to solve:
    1. Divide each by A: [ y/A = e^{kt} ]
    2. Apply natural logarithm: [ \ln(y/A) = kt ]
    3. Rearranging yields: [ t = \frac{\ln(y/A)}{k} ]
Example: Solve ( 100 = 20 e^{2t} )
  1. Divide both sides by 20: [ 5 = e^{2t} ]
  2. Logarithm application: [ \ln(5) = 2t ]
  3. Solve for ( t ): [ t = \frac{\ln(5)}{2} ]

Extraneous Solutions

  • While taking logarithms or solving, be cautious about extraneous solutions occurring from mathematical manipulations that may violate original equations’ conditions.
Example: Solving ( e^{2x} - e^{x} = 56 )
  1. Rewrite as a quadratic: ( (e^x)^2 - e^x - 56 = 0 )
  2. Factor and solve for valid solutions only, rejecting any non-real results.
Key Takeaway
  • Always check to ensure that solutions maintain the integrity of the original equations, especially with logarithmic forms where the input must be positive.