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ALGEBRA
 :: 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:

Common logarithms of xlog10 x (to the base 10) are often written log x, without the base explicitly indicated while, Natural logarithmsloge x (to the base e, where e = 2,718218... ), are written ln x.

Therefore,
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:
Solution:
Example:
Solution:
 :: Exponential and logarithmic equations
Exponential equations
In exponential equations the variable that has to be solved for is in the exponent.

To solve an exponential equation, rewrite the given equation to get all powers (exponentials) with the same base, or use logarithms when solving the exponential equation.

Examples:  a)  Solve  0.25x = 43x - 2    b)  Solve    43x + 2 = 64 22x - 4
         
                 42 43x = 43 22(x - 2) | 42
                        43x = 41 + x - 2
                        3x = x - 1
                        2x = - 1      =>   x = - 1/2.
Example:  Solve   4x + 2 + 4x + 1 + 4x -1  = 3x + 3 + 3x + 2.
Solution:
     
Example:   Solve     4x - 2 = 5x.
Solution:
Logarithmic equations

As logarithmic equations contain a logarithm of variable quantity, we use rules and properties of logarithms to solve a logarithm equation.

Example:   Solve    log x = -2.
Solution:    log x = log10-2,    x = 10-2 = 0.01.
Example:   Solve    log2 (log3 x) = 1.
Solution:                log3 = 21,     x = 32 = 9.
Example:  Solve    log (x + 5) -  log (2x - 3) = 2 log 2.
Solution:
                                
Example:  Solve   log (log x) + log (log x3 - 2) = 0.
Solution:
                 
 
 
 
 
 
 
 
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