Transcendental Functions Chapter 6

For x  0 and 0  a  1, y = log a x if and only if x = a y. The function given by f (x) = log a x is called the logarithmic function with base a. Every logarithmic equation has an equivalent exponential form: y = log a x is equivalent to x = a y A logarithmic function is the inverse function of an exponential function. Exponential function:y = a x Logarithmic function:y = log a x is equivalent to x = a y A logarithm is an exponent!

The function defined by f(x) = log e x = ln x is called the natural logarithm function. y = ln x (x  0, e  ) y x 5 –5 y = ln x is equivalent to e y = x In Calculus, we work almost exclusively with natural logarithms!

Examples

Derivative of Logarithmic Functions The derivative is Example: Solution: Notice that the derivative of expressions such as ln|f(x)| has no logarithm in the answer.

Example

Product Rule

Example