7/1/2015
Properties of Logarithms
 3 basic properties:
Ch. 3 – Exponential and Logarithmic
Functions
loga x + loga y = loga (xy)
loga x – loga y = loga (x/y)
3. loga xy = y loga x
1.
2.
3.3 – Properties of Logarithms
 Ex: Condense each logarithmic expression.
 Condense doesn’t mean evaluate!
 log 8 + log 12 – log 3
 log (8•12/3) =
log (32)
 2 ln(x + 2) – ln x
ln( x  2) 2  ln x  ln
Change of Base
Expand completely: log4 5x3y
 Ex: Evaluate log4 29.
 Set it equal to x, then scoop the loop!
 4x = 29
 log 4x = log 29
 x log 4 = log 29
 x = (log 29)/(log 4)
 x = 2.429
1. 3 log4 5x + log4 y
2. log4 5y + 3 log4 x
3. log4 5 + log4 x3 + log4 y
4. log4 5 + 3 log4 x + log4 y
1.
2.
ln
ln
( x  3) 2
x2  9
2.
4. ½ ln(4x + 1) – 2 ln x
3.
ln
4
ln
ln
ln
(4
x+
1)
–
2
2
1)
–
1)
+
4x
+
(4
x+
ln
n(
0%
ln
ln
x 3
x3
ln
2
x3
2x  6
x2  9
x
0%
x
0%
x
0%
2l
y
5•
3
lo
g4
x•
lo
g4
y
4y
0%
g4
og
lo
+l
lo
g4
x3
lo
g4
5+
3
lo
g4
5+
lo
g4
3. 2 ln(4x + 1) + 2 ln x
½
0%
2 ln( x  3)  ln( x 2  9)
1.
0%
0%
x+
3
5y
+
45
x+
og
lo
g4
3l
lo
lo
g4
g4
x
y
0%
Condense completely:
4x 1
x2
ln 4 x  1  2 ln x
ln 4 x  1  ln x 2
5. ln(4x + 1) – 4 ln x
0%
5. log4 5 • 3 log4 x • log4 y
change of base formula:
log x ln x log b x
log a x 


log a ln a log b a
lo
g4
 To evaluate non-base-10 logs on your calculator, use the
Expand completely:
( x  2) 2
x
4.
0%
0%
0%
0%
1
7/1/2015
Evaluate: log6 .01
1. -2.570
2. -1.285
3. -.333
4. 2.570
5. 1.285
0%
1.
28
5
0%
2.
57
0%
-.3
33
0%
-1
.2
85
-2
.5
7
0%
2