[FMQM] W00 - Powers, Logs and Exponents

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:
  • $\sqrt{x}=x^{\frac{1}{2}}$
  • $\sqrt{x}\times \sqrt{x}=x$
  • : 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

• $a{x}^2+bx+c=0$: 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: $(a^b{)}^c=a^{bc}$

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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

• $n={\log }_g\left(g^n\right)$

non-integer logs

• $a=lo{g}_{g}b$
  • $g^a=b$

log relations

• ${\log }_g(A\cdot B)={\log }_{g}A+{\log }_{g}B$
• $\ln \frac{1}{a}=-\ln \left(a\right)$

engineering application

• used to express power ratios
  • amplification factor
  • loss in communication channels
• $dB$ units:
  • $10lo{g}_{10}\left(ratio\right)$
• 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) $=10{\log }_{10}\left(100\right)=20dB$
    • for an output power that is 2 times input power
      • Gain (in dB) $=10{\log }_{10}\left(2\right)\cong 3dB$

change of logarithm base

• ${\log }_c\left(b\right){\log }_c\left(d\right)={\log }_d\left(b\right)$

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

• ${\log }_e=\ln$
• $\exp \left(x\right)\equiv e^x$: exponential notation
• $\exp \left(\ln \left(a\right)\right)=a$

• 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