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Understanding Exponentials and Logarithms

Oct 8, 2024

Lecture Notes: Exponentials and Logarithms

Exponential Equations

  • Key Principle: If two exponential expressions with the same base are equal, then their exponents are equal (i.e., if a^x = a^y, then x = y).

  • Solving Steps:

    1. Isolate the exponential expression.
    2. Take the logarithm of both sides using the law of logarithms.
    3. Use the power rule to bring the exponent out front.
    4. Solve for the variable.

  • Example:

    • Given equation: 3^(x+2) = 7
    • Steps:
      1. Take log of both sides.
      2. Bring (x + 2) out front using log rules.
      3. Solve: x = (log 7 / log 3) - 2.

  • Checking solutions:

    • Ensure precision by checking work, taking into account possible round-off errors.

Another Method: Substitution

  • Example:
    • Start with equation e^(2x) - e^x - 6 = 0.
    • Substitute: u = e^x, then solve the quadratic u^2 - u - 6 = 0.
    • Solutions: u = 3 or u = -2. Only u = 3 is valid (since exponential functions can't be negative).
    • Solve: e^x = 3, so x = ln(3).

Logarithmic Equations

  • Key Principle: If log base a of x equals log base a of y, then x = y.

  • Solving Steps:

    1. Use properties of logs to isolate the variable.
    2. Solve using exponential equivalences.

  • Example:

    • ln(x) = 8: Solution is x = e^8.
    • log base 2 (25 - x) = 3: Solution is x = 17.

Applications of Logarithms

  • Refraction of Light:
    • Light refraction changes its direction when passing through different mediums.
    • Beer-Lambert Law: Used to describe the intensity of light through a material.
    • Equation: I = Iā‚€ * e^(-k * x), where k is a constant for the material.
    • Example: Calculate light intensity at 20 feet with given parameters.

Compound Interest and Logarithms

  • Example Problem: Finding time to double an investment at a 5% interest rate.
  • Equations:
    • Semi-Annual Compounding: A = P(1 + r/n)^(nt)
    • Continuous Compounding: A = Pe^(rt)
  • Solution:
    • Semi-Annually: Takes ~14 years to double.
    • Continuously: Takes ~13.86 years.
  • Conclusion: Compounding more frequently has a small effect, but interest rate plays a more critical role in investment growth.

These notes summarize the concepts and examples discussed in the lecture on exponential and logarithmic equations, their solutions, and real-world applications through light refraction and compound interest.

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