Transcript 9.4 Properties of Logarithms Since a logarithmic function is the inverse of an exponential function, the properties can be derived from the.

9.4 Properties of Logarithms
Since a logarithmic function is the inverse of an exponential function,
the properties can be derived from the properties of exponents.
Properties of Logarithms
If M, N, & b are positive real numbers, b ≠ 1 and r is any real number,
then
logb MN  logb M  logb N
M
logb
 logb M  logb N
N
logb M r  r logb M
Ex 1) Derive: log b MN  log b M  log b N
Let w  log b M and
v  log b N
bw  M
bv  N
and
M  N  b w  bv
M  N  b w v
log b MN  log b b w v
 wv
 log b M  log b N
Using properties of logs, we can either expand or condense a
logarithmic expression.
Ex 2) Express in expanded form.
x  y

a) ln
x4 y
1
2
 ln  x  y   ln x 4  ln y
1
2
 12 ln( x  y )  4 ln x  ln y
b) Make up your own expression using log4, p, q, and r.
Write your problem on a whiteboard
Give your whiteboard to someone close. Solve theirs.
Return your board & evaluate their work.
Ex 3) Express in condensed form.
a)
1
3
log x 2  12 log y 3  log 3 x 2  log y 3  log
b) Make another of your own using ln, x, & y.
Trade & solve again.
3
x2
y3
Solving log equations
Note: logb x = logb y
iff
x=y
Also: Often, you will need to condense logs first in order to solve
Ex 4) Determine the domain. Then solve.
log5 (x + 3) – log5 (x – 2) = 1
must be positive
x + 3 > 0 and x – 2 > 0
x > –3 and
x>2
Solving: log x  3  1
5
x2
x3
1
5 
x2
D: {x: x > 2}
5 x  10  x  3
4 x  13
13
x
4
Real world formulas often must be rearranged to isolate the variable
you are solving for.
Ex 5) Formula relating height (h), in miles, and atmospheric pressure
(P), in pounds per in2 is:
(ln P  ln14.7)
h
0.21
0.21h  (ln P  ln14.7)
P
 0.21h  ln
14.7
P
0.21h
e

14.7
14.7e0.21h  P
Solve for P
Homework
#905 Pg 464 #1–43 odd, 2, 18, 34, 36, 44