Chapter 5 Logarithmic Exponential and Other Transcendental Functions

Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions

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

y The function defined by f(x) = loge x = ln x (x 0, e 2. 718281 ) is called the natural logarithm function. y = ln x 5 – 5 y = ln x is equivalent to e y = x In Calculus, we work almost exclusively with natural logarithms! x

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

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 statement is a solution to the.

4. Show that statement is a solution to the.

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

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