Properties of Logarithms

• Definition of logs:
  y = log _a x if and only if (iff) a^y = x    a > 0;   a ≠ 1;   x ≠ 0
  example:      2 = log ₁₀ 100    iff   10 ² = 100


• log ₁₀ 1 = 0  iff  10⁰ = 1 Therefore log _a 1 = 0  This states that a log 1 to any
  base equals 0 because the law of exponents require that a⁰ = 1.


• log ₁₀ 100 = 2  since 100 = 10² we can substitute 10² for 100 in log ₁₀ 100 → log ₁₀ 10² → log ₁₀ 10² = 2  
  This leads us to the general property:  log _a a^t = t
  So what does ln e² = ? answer



• log ₁₀ 1000 = 3 and 10³ = 1000 so lets substitute log ₁₀ 1000 for the exponent 3 which leads to   10^{log ₁₀ 1000} = 1000
  That gives us the general property   a^log _a t = t
  What does e^ln  43 = ? answer
  


• log _a r = log _a t   iff   r = t
  


• B^t = B^u   iff   t = u


• log_a uv = log_a u + log_a v
  ex. log (12)(14) = log 12 + log 14 = ? answer
  


• log_a( u/v) = log_a u – log_a v
  ex. log (12/14) = log 12 – log 14 = ?answer
  


• log_a uⁿ = nlog_a u
  ex. log 12³ = 3 log 12 = ? answer
  


• THE ABOVE PROPERTIES APPLY TO ALL LOGARITHMS INCLUDING ln WHICH ARE JUST LOGS BASE e
  

return
return PCTC