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)

2. Doubling

   A = A₀(2)ᵗ/ᵈ

   • A: final quantity

   • A₀ : initial quantity

   • t: time passed

   • d: doubling time (how long it takes to double)

3. 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)