
A logarithm is the functional inverse of exponentiation. Written in algebraic notation,
The exponent, x in the equation, is also called the antilogarithm. In manipulative 1, Note that the value of x and y on the point on the blue line (y = e^{x}), is swapped on the point on the green line (y = log_{e}x). This is a consequence of y = log_{e} x being the inverse of y = e^{x}. 
To convert an exponent to a logarithm, use the definition of logarithms:
log_{a} y = x iff y = a^{x}.
Start with 5^{x} = 25. Substitute like values into the definition.
5^{x} = 25 ⇒ x = log_{5} 25.
To convert a logarithm to an exponent, use the definition of logarithms:
log_{a} y = x iff y = a^{x}.
Start with log_{2} x = 3. Substitute like values into the definition.
log_{2} x = 3 ⇒ x = 2^{3}.
An equation such as 3^{x} = 5 is difficult to solve without using logarithms. However, converting the equation to a logarithm makes it easy to solve. Use the definition of a logarithm: 3^{x} = 5 ⇒ log_{3} 5 = x. Calculating the logarithm gives x ≈ 1.46497.
The common logarithm is a logarithm using a base 10. When writing a common logarithm, leave out the base: log 14 means the same thing as log_{10} 14. Common logarithms are commonly used in business applications.
The natural logarithm is a logarithm with a base e. The natural logarithm is written ln x. ln x means the same thing as log_{e} x. Natural logarithms are used in mathematics and science.
Name  Equation 

Product Property  
Quotient Property  
Power Property  
Change of Base Property  
Table 1: Properties of Logarithms 
Click on the blue points on the sliders and drag them to change the figure What happens when the base (b) is set to 1? Why? 
Manipulative 2  Graph of a Logarithmic Function Created with GeoGebra. 
#  A  B  C  D 
E  F  G  H  I 
J  K  L  M  N 
O  P  Q  R  S 
T  U  V  W  X 
Y  Z 
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