Download 2.4 Exponetial and Logarithmic Functions Exponential Functions

2.4
Exponetial and Logarithmic Functions
Exponential Functions
Laws of Exponents
If x, y, a, and b are real numbers, with a > 0 and b > 0, then
1.
ax ▪ ay = ax+y
2.
4.
1x
a-x
=1
5.
(ax)y = axy
1
1
= x =  
a
 a
3.
(ab)x = ax ▪ bx
6.
a0 = 1
x
An exponential function is a function of the form
f(x) = bx
where b > 0 and b ≠ 1. The domain of f is the set of all real numbers.
Example: Graph f(x) = 2x and
1
f(x) = 2-x =  
2
x
Properties of the Graph of y = bx
1.
The domain is all real numbers, and the range is all positive real
numbers.
2.
There are no x-intercepts, and the y-intercept is (0,1).
3.
The x-axis is horizontal asymptote.
4.
If b > 1, then f(x) is an increasing function and is one-to-one.
5.
If 0 < b < 1, then f(x) is a decreasing function and is one-to-one.
Exponential Equations
Since exponential functions are one-to-one functions, the following
property is true:
If ax = ay , then x = y
Example: Solve for x
27
x–1
= 9 2x – 3
Example: Solve
3 x 4 x+(1/2) = 3,456
The Base e
The number e is defined as
e = lim 1 
n 

1 n
n
≈ 2.7182818...
The natural exponential function f is defined by
f(x) = ex
for every real number x.
Logarithmic Functions
The word “logarithm” means “exponent.”
y = log b x
logarithmic
form
if and only if
x = by
exponential
form
Since b > 0 and b ≠ 1, then x > 0.
Example: Graph y = 2x and y = log 2 x
Properties of the Graph of f(x) = log b x
1.
2.
3.
4.
The domain is all positive real numbers, and the range is all real
numbers.
The x-intercept of the graph is 1. There is no y-intercept.
The y axis is a vertical asymptote of the graph.
The graph is decreasing if 0 < b < 1 and increasing if b > 1.
Common Logarithm
Common logarithm refers to log base 10, and is written without the base.
log 10 x = log x
The log button on a calculator is the common logarithm.
Natural Logarithm
Natural log is logarithm to the base e, where
e = 2.7182818.... Special notation
log e x = ln x.
The ln button on a calculator is natural logarithm.
Properties of Logarithms
1.
log a 1 = 0
2.
log a a = 1
3.
log a ax = x
4.
a log a x = x
5.
log b xy = log b x + log b y
6.
log b
7.
x
= log b x – log b y
y
log b x p = p log b x
8.
log b x = log b y iff x = y
Change of Base Formula
log a x =
Example: Solve
log x ln x

log a ln a
log 2 (x + 3) – log 2 (x – 6) = log 2 (x – 5)
Example: Solve log 3 (2x + 5) + log 3 (x – 1) = 2
Example: Solve
Example: Solve
7x+2 = 43x
ex+3 = x