Complex Analysis/Elementary Functions/Logarithmic Functions

Logarithmic Functions Logarithm A logarithm is the exponent that a base is raised to get a value. Such exponential equations can be written as logarithmic equations and vice versa. Exponential equations are in the form of bx = a , and logarithmic equations are in the form of logba = x . When converting from exponential to logarithmic form, and vice versa, there are some key points to keep in mind: 1. The base of the exponent become the base of the logarithm. Example: 37 = 2187 log32187 = 7 2. The exponent is the logarithm. Example: 52 = 25 log525 = 2 3. Any nonzero base to the 0 power is 1. 60 = 1 log61 = 0 4. An exponent or log can be negative. 4-2 = 0.0625 log40.0625 = -2 5. The exponent and the log can be variables. 4y = 1024 log41024 = y A logarithm is also an exponent. This means that the exponent rules apply to logarithms as well. A common logarithm is a logarithm that has a base of 10. Bases of logarithms are known to be 10 when there is no base written for them. For example: log6 = log106 Logarithmic functions are inverses of exponential functions, since logarithms are inverses of exponents. For example: y = 3x is the inverse of y = log3x And, since these two functions are inverses, their domain and ranges are switched. So, for y = 3x the domain is all real numbers and the range is y > 0. And, for y = log3x the domain is x > 0 and the range is all real numbers.