Download Section 7.4* - Gordon State College

Section 7.4*
General Logarithmic and
Exponential Functions
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.
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
LAWS OF EXPONENTS
If x and y are real numbers and a, b > 0, then
1. a
x y
2. a
a a
x
x
a
 y
a
x y
 
3. a
y
x y
a
xy
4. ( ab)  a b
x
x
x
DIFFERENTIATION OF GENERAL
EXPONENTIAL FUNCTIONS
 
d x
x
a  a ln a
dx
d g ( x)
g ( x)
a
 a ln a g ( x)
dx


ANTIDERIVATIVES OF GENERAL
EXPONENTIAL FUNCTIONS
x
a
 a dx  ln a  C a  1
f ( x)
a
f ( x)

f
(
x
)
a
dx


C
a

1

ln a
x
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
log a x  y if, and only if,
a x
y
NOTES ON THE GENERAL
LOGARITHMIC FUNCTION
1. loge x = ln x
2. a
loga x
 
 x, log a a  x
x
THE CHANGE OF BASE
FORMULA
For any positive number a (a ≠ 1), we have
ln x
log a x 
ln a
DIFFERENTIATION OF GENERAL
LOGARITHMIC FUNCTIONS
d
1
 log a x  
dx
x ln a
d
g ( x)
 log a g ( x)  
dx
g ( x) ln a
THE GENERALIZED VERSION
OF THE POWER RULE
Theorem: If n is any real number and f (x) = xn,
then
n 1

f ( x)  n x
THE NUMBER e AS A LIMIT
e  lim (1  x)
1/ x
x 0
 1
 lim 1  
n
 n
n