y = 10^{ x}For example it is obvious that the number 1000 can be expressed as 10^{ 3}, because the exponent 3 means to multiply 10 by itself 3 times and 10·10·10=1000. It is not so obvious that the number 16 can be expressed as 10^{ 1.2}. Clearly this cannot mean to multiply 10 by itself 1.2 times! What does it mean and how can we calculate that this is the correct power of 10? We start by giving it a name: logarithm. We say that 1.2 is the logarithm of 16. It is the subject of this chapter.
Definition: The logarithm is the function that takes any positive number x as input and returns the exponent to which the base 10 must be raised to obtain x. It is denoted log(x). |
log ( 10^{ 5.7} ) = 5.7
log ( 1000 ) = log ( 10^{ 3} ) = 3
log ( 16 ) = 1.2
Guided by these examples, we now give the following, more general definition of a logarithm in any base:
16 = 2^{ 4} → log_{ 2 }(16) = 4 16 = 4^{ 2} → log_{ 4 }(16) = 2 16 = 16^{ 1} → log_{ 16 }(16) = 1
Definition: The logarithm to base b is the function that takes any positive number x as input and returns the exponent to which the base b must be raised to obtain x. We denote it as log_{b}(x). |
x = b^{ y}.This says that ‘x is b raised to the exponent y’. This is called the exponential form of this relationship. But this same relationship can also be written as
log_{ b}(x) = y,This says that ‘the logarithm of x to base b is y’. This is called the logarithmic form of this relationship. These two equations are equivalent. They are like saying ‘Mary is John’s mother’ and ‘John is Mary’s son’. We say that we are ‘taking logs’ when converting from exponential to logarithmic form, and ‘taking antilogs’ when converting from logarithmic to exponential form.
log_{ b}( b^{ y }) = y,and if we substitute the logarithmic form log_{ b}(x) = y into its own exponential form b^{ y} = x we get the identity These identities show how the inverse operations of logging and antilogging undo each other.
log_{10}(10^{ 1.2} ) = 1.2,and
Multiplication property Division property Exponentiation property
b^{ m} = x and b^{ n} = yNote for later reference that these substitutions expressed in logarithmic form are:
m = log_{ b}(x) and n = log_{ b}(y).With the substitutions the three properties of exponents read: Now take logs of these three equations (i.e. write them in logarithmic form): Now substitute log_{ b}(x) for m and log_{ b}(y) for n on the right side of each property (but only for m in the third one). The result is three properties of logarithms.
Common logarithms Natural logarithms the base 10 e ≈ 2.718 an example in exponential form y = 10^{ 3} = 1000 y = e^{ 3} ≈ 20.08 the same example in logarithmic form log(y) = 3 ln(y) = 3
Logarithmic form of the Change of Base Formula: Exponential form of the Change of Base Formula: These formulae are used to convert from an inconvenient base c to a convenient base b (usually 10 or e). The left side is usually converted to the right side. The quantity log_{ b}(c) is the conversion factor. |
7 = e^{ ln (7)} = e^{ 1.946}Substitute this expression in for 7 in y = 7^{ x}.
y = (7)^{ x} = (e^{ 1.946 })^{ x} = e^{ 1.946 x}
Algebra Coach Exercises |