Logarithmic, Exponential, and Other Transcendental Functions Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

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

In Calculus, we work almost exclusively with natural logarithms! 5 –5 y = ln x The function defined by f(x) = loge x = ln x (x  0, e 2.718281) is called the natural logarithm function. y = ln x is equivalent to e y = x In Calculus, we work almost exclusively with natural logarithms!

Definition of the Natural Logarithmic Function

Theorem 5.1 Properties of the Natural Logarithmic Function

Natural Log

Natural Log Passes through (1,0) and (e,1). You can’t take the log of zero or a negative. (Same graph 1 unit right)

Theorem 5.2 Logarithmic Properties

Properties of Natural Log: Expand: Write as a single log:

Properties of Natural Log: Expand: Write as a single log:

Definition of e

Theorem 5.3 Derivative of the Natural Logarithmic Function

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

Example

Example

Example Product Rule

Example

Example

Example

Example

Theorem:

Theorem:

Theorem 5.4 Derivative Involving Absolute Value

Try Logarithmic Differentiation.

4. Show that is a solution to the statement .

4. Show that is a solution to the statement .

At (1, 3) the slope of the tangent is 2 Find the equation of the line tangent to: at (1, 3) At (1, 3) the slope of the tangent is 2

Find the equation of the tangent line to the graph of the function at the point (1, 6).