1. General Logarithmic and Exponential Functions
Section 7.4*
General Logarithmic and Exponential Functions

2. GENERAL EXPONENTIAL FUNCTIONS
Definition: If a > 0, we define the general exponential function with base a by
f (x) = ax = ex ln a
for all real numbers x.

3. NOTES ON f(x) = ax 1. f (x) = ax is positive for all x
2. For any real number r, ln (ar) = r ln a

4. LAWS OF EXPONENTS
If x and y are real numbers and a, b > 0, then

5. DIFFERENTIATION OF GENERAL EXPONENTIAL FUNCTIONS

6. ANTIDERIVATIVES OF GENERAL EXPONENTIAL FUNCTIONS

7. THE GENERAL LOGARITHMIC FUNCTION
Definition: If a > 0 and a ≠ 1, we define the logarithmic function with base a, denoted by loga, to be the inverse of f (x) = ax. Thus

8. NOTES ON THE GENERAL LOGARITHMIC FUNCTION
1. loge x = ln x 2.

9. THE CHANGE OF BASE FORMULA
For any positive number a (a ≠ 1), we have

10. DIFFERENTIATION OF GENERAL LOGARITHMIC FUNCTIONS

11. THE GENERALIZED VERSION OF THE POWER RULE
Theorem: If n is any real number and f (x) = xn, then

12. THE NUMBER e AS A LIMIT