

Exponential and Logarithmic
Functions and Equations

Rules and properties of logarithms

Natural logarithm, common logarithm

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



Notation:
Common logarithms of x,
log_{10} x
(to the base 10) are often written log
x, without the base
explicitly indicated while, Natural
logarithms, log_{e}
x (to the base e,
where e = 2,718218... ), are written ln
x. 
Therefore, 
log_{a}
a^{x} =
x 

log_{a}
a = 1

log_{a} 1 = 0


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, 


and 




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


Example:
If a
= log _{10 }2 then
log _{10 }25
= ?. 


Example:
Find the value of the expression 




Example:
Find the value of the expression 




Example:
Find the value of the expression 




Example:
If a
= log 5 and b = log 3
then log
_{30 }8 = ?. 




Example:
Find the value of the expression 











College
algebra contents D 



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