Precalculus Exponential and Logarithmic Functions
Properties of Logarithms
   Concepts 1

Statement 1
.
Logarithm - of - a - Product Property
log a (x·y) = loga x + loga y

Example 1. Using log103 = 0.4771 and log105 = 0.6990 , compute log1015 .

Solution
log 10 15 = log10 (3·5)
  = log10 3 + log10 5
  = 0.4771 + 0.6990
  = 1.1761 .
Answer:   log 10 15 = 1.1761 .

Statement 2
.
Logarithm - of - a - Quotient Property


Example 2. Using log103 = 0.4771 , compute .

Solution
  =
  = log10 10 - log10 3
  = 1 - 0.4771
  = 0.5229 .
Answer:   = 0.5229 .

Statement 3
.
Logarithm - of - a - Power Property
loga (xb) = b·loga x

Example 3. Using log102 = 0.3010 , compute log1016 .

Solution
log10 16 = log10 (24)
  = 4·log102
  = 4·0.3010
  = 1.2040 .
Answer:   log10 16= 1.2040 .

Statement 4
.
Change of Base Formula

Example 4. Using ln10 = 2.3026 and ln5 = 1.6094 , compute log105 .

Solution
log105 =
  =
  = 0.6990 .
Answer:   log 10 5 = 0.6990 .


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