# Logarithms

A logarithm is an exponent.  A logarithm is an exponent which indicates to what power a base must be raised to produce a given number.

y = bx             exponential form

x = logb y        logarithmic form

x is the logarithm of y to the base b

logby is the power to which we have to raise b to get y

We are expressing x in terms of y

Examples

x = logb y

 x = log2 8 This means the logarithm of 8 to the base 2.  It is the exponent to which 2 must be raised to get 8.  We know that 2(2)(2) = 8.  Therefore x = 3.

 x = log6 36 This means the logarithm of 36 to the base 6.  It is the exponent to which 6 must be raised to get 36.  We know that 6(6) = 36.  Therefore x = 2.

 x = log10 10,000 This means the logarithm of 10,000 to the base 10.  It is the exponent to which 10 must be raised to get 10,000.  We know that 10(10)(10)(10) = 10,000.  Therefore x = 4.

 log b b  =  1 The logarithm of any number to the same base equals 1.

 x  =  log 11 11 This means the logarithm of 11 to the base 11. It is the exponent to which 11 must be raised to get 11.  We know that 1 (1) = 11.  Therefore x = 1.

 log b 1  = 0 The logarithm of 1 always equals 0.

Any number can serve as b, the base.

### Common (Briggsian) logarithms  The base is 10.

Logarithms to the base 10 are widely used.  Thus it is common to drop the subscript. If the base does not appear it is understood that the base is 10.

log10 y = log y

### Natural (Naperian) logarithms      The base is e.

Remember e is the irrational number where e = 2.71828...  The symbol "ln" refers to natural logarithms.

 loge x = ln x ln x is the exponent to which e must be raised to get x.

Why do we want to use logarithms?  To simplify calculations in many cases.

### Rules for logarithms

 Product rule Quotient rule Power rule This rule is useful because it allows us to solve equations where the variable is an exponent.

Exponential and Logarithmic Functions are inverse functions

Consider the following tables and the associated graphs:

 x f(x) = ex x f(x)= ln x 0 1 1 0 1 2.7 2.7 1 2 7.39 7.39 2 3 20 20 3  [Index]