Exponential and Logarithmic Functions and Equations
      Translated logarithmic and exponential functions
         Rules and properties of logarithms
         Natural logarithm and common logarithm conversions
         Changing the base different logarithmic identities
Rules and properties of logarithms
A logarithm is the exponent (the power) to which a base must be raised to yield a given number, that is
y = loga if   x = a y.
Examples:  
 
Notation: Common logarithms of x, log10 x (to the base 10) are often written log x, without the base explicitly indicated while, Natural logarithms, loge x (to the base e, where e = 2,718218... ), are written ln x.
Therefore,
loga ax = x loga a = 1 loga 1 = 0
Logarithms are used to simplify multiplication, division and exponentiation so that,
   loga (m n) = loga m + loga n    
  loga mn = n loga m    
Example:  Using the rules of logarithms find the value of x.
 Solution:    
 
Changing the base different logarithmic identities
Using the identity

In a similar way

therefore,    

Similarly,
and

Example:   If   log 2 x = 7  then  log 4 2x = ?.
Solution:
Example:   If   a = log 10 then  log 10 25 = ?.
Solution:
Example:   Find the value of the expression
Solution:
Example:   Find the value of the expression
Solution:
Example:   Find the value of the expression
Solution:
Example:    If   a = log 5 and  b = log 3  then  log 30 8 = ?.
Solution:
Example:   Find the value of the expression
Solution:
Pre-calculus contents F
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