12-07: Logarithms (II)

Drop the Base

You can solve same bases’ exponents by dropping the bases

  • May need to make things into the same base

If you can, make the bases the same as a power of the smallest base if possible (power of a power rule)

Exponential Growth and Decay

  1. % Growth/Decay

   > A = A₀(1± r)

  • A: final quantity
  • A₀: initial quantity
  • r: growth/decay rate (express as a decimal)
  1. Doubling

   > A = A₀(2)ᵗ/ᵈ

  • A: final quantity
  • A₀ : initial quantity
  • t: time passed
  • d: doubling time (how long it takes to double)
  1. Half life

   > A = A₀(1/2) ᵗ/ʰ

  • A: final quantity
  • A₀ : initial quantity
  • t: time passed
  • h: half life (how long it takes to half in quantity

Note: time units must be the same

Product Law of Logs

Quotient Law of Logs

Techniques for Solving Log Equations

  • If just a single log term on either side of the equal sign → cancel logs

   

  • If both log terms and constants in an equation → rearrange, logs on one side and constants on the other
  • When +/- log terms → combine with product and quotient laws (above)
  • Change to exponential form:

 

  • Check for extraneous roots
    • ==Extraneous root==: answer that throws error when plugged into the question
    • Cannot log 0 or a negative numbers → sub answers back in to check
    • Should be greater than 0 when checking (make expression >0 when checking and if it works then it doesn’t break the domain)