::
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 =
log_{a}
x
if x = a_{
}^{y}. 


Examples: 






Logarithms
are used to simplify multiplication, division and exponentiation
so that,

log_{a}
(m · n) =
log_{a}
m +
log_{a}
n


log_{a}
m^{n} =
n ·
log_{a}
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, 




Example: 
If log _{2
}x
= 7 then log
_{4} 2x = ?. 



Example: 

Solution: 




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 0.25^{x}
= 4^{3x}^{
 }^{2} 
b) Solve 4^{3x}^{
}^{+ }^{2}
= 64 · 2^{2}^{x}^{
 }^{4} 

4^{2}
· 4^{3x}
= 4^{3} · 2^{2(}^{x}^{
 }^{2)}^{
} ¸
4^{2} 
4^{3x}
= 4^{1 + }^{x}^{
 }^{2} 
3x = x
 1 
2x
=  1
=>
x
=  1/2. 



Example:
Solve 4^{x}^{
}^{+ }^{2}
+ 4^{x}^{
}^{+ }^{1
}+ 4^{x}^{
}^{1}
= 3^{x}^{
+ }^{3}
+ 3^{x}^{
+ }^{2}.




Example:
Solve
4^{x}^{
}^{
2} =
5^{x}.



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 log_{2}
(log_{3} x) = 1. 
Solution:
log_{3} x
= 2^{1},
x = 3^{2}
= 9. 


Example:
Solve log
(x + 5) 
log
(2x 
3) = 2 · log 2.




Example:
Solve log
(log
x) + log
(log
x^{3} 
2) = 0.



