power

  • \( 2 \times 2 \times 2 = 2^3 \)
    • \(3\): exponent
    • \(2\): base
  • \( 2^0 = 1 \)
  • \( 2^{-1} = \frac{1}{2} \) : reciprocal

  • \( 2^{-2} = \frac{1}{2^2} = \frac{1}{4}\)

  • generalizing:
    • \( x^0 = 1 \)
    • \( x^{-a} = \frac{1}{x^a} \)
  • multiplication
    • \( 2^3 \times 2 = 2^4 \)
  • division
    • \( \frac{2^3}{2} = 2^2 \)
  • reciprocal
    • \( \frac{1}{x} = x^{-1} \)
    • explodes at origin (origin singularity)

squares and square roots

  • square:
    • \( x^2 \): gives area of square of side \(x\)
  • square root:
    • \( \sqrt{x} = x^{\frac{1}{2}} \)
    • \( \sqrt{x} \times \sqrt{x} = x \)
    • \( \sqrt{} \): radical
  • both \( -2 \) and \( 2 \) are square roots of 4
    • \( \sqrt{2} \times \sqrt{2} = 4 = \sqrt{-2} \times \sqrt{-2} \)
  • if the square root is a distance
    • like in the pythagorean theorem
    • only the positive is considered

quadratics and roots

  • \( ax^2 + bx + c = 0\): quadratic equation
    • roots are \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • roots are point where the plot of the equation cross the x-axis
  • here the root is a solution to an equation
    • then both negative and positive square roots have to be considered

powers of powers

  • \( (2^3)^2 = (2 \times 2 \times 2) \times ( 2 \times 2 \times 2) = 2^6 \)
  • generalizing: \( (a^b)^c = a^{bc}\)

logs and exponentials

  • logarithm is inverse power operation
  • a log operation has two inputs
    • the base
    • a number to be operated on
  • without a base specified a log operation is meaningless
  • example: “log to the base 2 of 8 is 3”
    • \( \log_2{8} = 3 \)
    • which means \( 2^3 = 8 \)

integer logs

  • \( n = \log_g{(g^n)} \)

non-integer logs

  • \( a = log_g{b} \)
    • \( g^a = b \)

log relations

  • \( \log_g{(A\cdot B)} = \log_g{A} + \log_g{B} \)
  • \( \ln{\frac{1}{a}} = -\ln{(a)} \)

engineering application

  • used to express power ratios
    • amplification factor
    • loss in communication channels
  • \(dB \) units:
    • \( 10 \text{ } log_{10}{(ratio)} \)
  • example:
    • to compute amplifier gain
      • Gain (in dB) \( = 10 \log_{10}{\left( \frac{P_{out}}{P_{in}} \right)} \)
      • for an output power that is 100 times input power
        • Gain (in dB) \( = 10 \log_{10}{(100)} = 20 dB \)
      • for an output power that is 2 times input power
        • Gain (in dB) \( = 10 \log_{10}{(2)} \cong 3 dB \)

change of logarithm base

  • \( \log_c{(b)}\log_c{(d)} = \log_d{(b)} \)

common bases for log

  • \( \log_{10} \):
    • engineering applications and calculators
    • 10 finger counting and decimal system
  • \( \log_2 \):
    • computer science with binary counting
  • \( \log_{e} \):
    • \( e \): base of the natural logarithms
    • fundamental physics and mathematics \(\log\)

exponential and natural logarithm

  • \( \log_e{} = \ln \)
  • \( \exp{(x)} \equiv e^x \): exponential notation
  • \( \exp{(\ln{(a)})} = a\)

exp-ln

  • exponential function:
    • for larger negative arguments,
      • gets closer and closer to the x-axis
    • for larger positive arguments
      • grows faster and faster to the x-axis
  • log functions:
    • for smaller positive arguments
      • arbitrarily large and negative