Transcript 8 – 4 Logarithmic Functions Day 1

8 – 4 : Logarithmic Functions
(Day 1)
Objective: Be able to evaluate
Logarithmic Functions.
You know that 2  4 and 2  8. However,
2
3
for what value of x does 2  6? It is reasonalbe
to assume that x must lie between 2 and 3.
To find the exact value of x, mathematicans defined
logarithms.
x
Definition of Logarithm with Base b
Let b and y be positive numbers b ≠ 1.
The logarithm of y base b is denoted by
Logby and is defined as follows:
logb y  x if and only if b  y
x
The expression logb y is read "log base b of y."
log b y  x and b  y
x
logarithmic form
exponential form
Example 1: Rewriting Logarithmic
Expressions
A.) log232 = 5
25 = 32
B.) log51 = 0
50 = 1
C.) log101 = 1
101 = 10
D.) log10 0.1 = -1
10-1 = 0.1
E.) log1/2 2 = -1
(1/2)-1 = 2
Special Logarithm Values
Let b be a positive real number such
that b ≠ 1
Logarithm of 1
log b1  0 because b  1
0
Logarithm of Base b logb b  1 because b  b
1
Example 2: Evaluating Logarithmic
Expressions
Evaluate the expression.
a. log3 81  x
3x = 81
3 raised to what power equals 81?
3  81 log 3 81  4
4
b. log 5 0.04  x
5x = 0.04
5 raised to what power equals 0.04?
1
1
5  2 
 0.04
5
25
 log 5 0.04  2
2
c.
log 1 8
2
½ raised to what power equals 8?
3
1
   8  log 1 8  3
2
2
d. log 9 3
9 raised to what power equals 3?
1
2
1
9  3 log 9 3 
2
Common Logarithm Natural Logarithm
log10 x  log x
log e x  ln x
ln and e are inverses of each other
Example 3: Evaluating Common and
Natural Logarithms
Expression
a. log 5
b. ln 0.1
Keystrokes
Display
LOG 5 ENTER 0.69870
LN 0.1
-2.302585
Homework:
P. 490 #16 – 34 even, 36 -47 all
Example 4: Evaluating a Logarithmic Function
The slope s of a beach is related to the average
diameter d (in millimeters) of the sand particles
on the beach by this equation.
s  0.159  0.118log d
Find the slope of a beach if the average diameter
of the sand particles is 0.25 millimeters.
Solution
If d = 0.25, then the slope of the beach is:
s  0.159  0.118log d
 0.159  0.118log 0.25
 0.159  0.118  0.602 
 0.159  0.071
 0.09