Name: ________________________________________
Unit 5 Lesson 7 Do Now (Do Not Hand In This Do Now)
1.
Find the exact value of the logarithmic expression without using a graphing calculator .
(a) log
2
8
3
(b) log ç
æ
è
5
500
ö
ø
÷ (c) ln
1
4 e
3
(d) ln e
ln e
2
2.
Use properties of logarithms to write the expression as a single logarithm .
4 ln x
2 y
+
3ln yz
3
3. Use properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. Assume all variables are positive. log x
2
( x
+
5)

Pre-Calculus Honors
Book Reference 3.5
Unit 5 Lesson 7: Solving Exponential and Logarithmic Equations (2 Day Lesson)
Objective: ______________________________________________________________
Method # 1: Using a common base to solve exponential equations.
Guided Practice Example: Mark up and write out the steps, in your own words, to the following example.
 
( 3 x

4 ) 
1
4
Step 1:
( ) 2
·
(3 x
+
4) =
(4)
-
1
Step 2:
( x
+
4
) = -
1
Step 3:
6 x
+
8
= -
1
Step 1:
_____________________________________________
_____________________________________________
Step 2:
_____________________________________________
_____________________________________________
Step 3:
_____________________________________________
_____________________________________________
6 x
= -
9 x
= -
9
6
= -
3
2
Try these examples within your groups…
(a) 27
( x
+
3) =
9
(1
x )
(b) 5
2
x = ç
æ
è
1
125
÷
ö
ø
9 x

Method #2: Using logarithms to solve exponential equations.
Guided Practice Example: Mark up and write out the steps, in your own words, to the following example.
7
-
3 e
-
2 x =
2
Step 1:
-
3 e
-
2 x = -
5
Step 1:
_____________________________________________
_____________________________________________
Step 2: e
-
2 x =
5
3
Step 3: ln
( )
= ln
ç
æ
è
5
3
ö
ø
÷
Step 2:
_____________________________________________
_____________________________________________
Step 3:
_____________________________________________
_____________________________________________
Step 4:
-
2 x
· ln
( ) = ln ç
æ
è
5
3
ö
ø
÷
Step 5:
-
2 x
= ln ç
æ
è
5
3
ö
ø
÷
Step 4:
_____________________________________________
_____________________________________________
Step 5:
_____________________________________________
_____________________________________________
Step 6: x
= ln ç
æ
è
-
2
5
3
÷
ö
ø
» -
0.2554
Step 6:
_____________________________________________
_____________________________________________
Try these examples within your groups…Round answers to three decimal places.
(a) 3 7 x

0 .
2 (b) 40 e 0 .
6 x 
3

237
(c) 3 e
4 x =
45 (d) 0.75
e
3.4
x -
0.3
=
80.1

Method #3: Using logarithms to solve exponential equations with different bases.
Guided Practice Example: Mark up and write out the steps, in your own words, to the following example.
4 x

1 
27 x

1
Step 1:
( ) x
-
1 = ( ) x
+
1
Step 1:
_____________________________________________
_____________________________________________
Step 2:
Step 2: ( x
-
1)
· ( ) =
( x
+
1)
· ( )
_____________________________________________
_____________________________________________
Step 3: 1.3863
·
( x
-
1)
=
3.2958
·
( x
+
1)
Step 3:
_____________________________________________
_____________________________________________
Step 4:
Step 4:
1.3863
x
-
1.3863
=
3.2958
x
+
3.2958
_____________________________________________
_____________________________________________
-
4.6821
=
1.9095
x x
» -
2.452
Try these examples within your groups…Round answers to three decimal places.
(d) 5
2 x

3 
3 x

1 b.) 6
x
-
2 =
9
x
-
1
Challenge: e
2 x +
6 e x =
16

Name: _________________________________________________
Pre-Calculus Honors Unit 5 Lesson 7 (Day 2): Do Now
1.
Find the domain of the function without using a graphing calculator. f ( x )
= log ç
è
æ
-
1
(5
-
2 x )
ö
ø
÷
2.
Find the exact value of the logarithmic expression without using a graphing or scientific calculator .
(a) log
4
3
192
(b) log
2
1
5 8
(c) 0.2 ln e
-
4 + ln e
2
3. Solve the following equation. Round your final answer to three decimal places .
-
2
-
1.8
2 x
= -
11.6
Name: _________________________________________________
Pre-Calculus Honors Unit 5 Lesson 7 (Day 2): Do Now
1.
Find the domain of the function without using a graphing calculator. f ( x )
= log ç
è
æ
-
1
(5
-
2 x )
ö
ø
÷
2.
Find the exact value of the logarithmic expression without using a graphing or scientific calculator .
(a) log
4
3
192
(b) log
2
1
5 8
(c) 0.2 ln e
-
4 + ln e
2
3. Solve the following equation. Round your final answer to three decimal places .
-
2
-
1.8
2 x
= -
11.6

Day 2 : 3. Solving logarithmic equations.
Guided Practice: Solving Logarithmic Equations Algebraically log
6
( x

5 )
 log
6 x

2
Step 1: log
6 x ( x
+
5)
=
2
Step 2:
( ) log
6 x ( x
+
5) = ( ) 2
Step 3: x ( x
+
5)
=
36
Step 4: x 2
+
5 x
=
36
Step 1:
_____________________________________________
_____________________________________________
Step 2:
_____________________________________________
_____________________________________________
Step 3:
_____________________________________________
_____________________________________________
Step 4:
_____________________________________________
_____________________________________________ x 2
+
5 x
-
36
=
0
( x
+
9)( x
-
4)
=
0 x
= -
9( extraneous ) x
=
4

Try these examples within your groups…Round answers to three decimal places.
(1) ln( 2 x )

12 (2) 3
 log( x

2 )

5
(3) log( x

3 )
 log x

1 (4) 2 log
3
( x

4 )
 log
3
9

2
(5) log
2
( x

1 )
 log
2
( x

1 )

3
(6) log 4 x
log
4
( x
-
1)
=
1
2

(7) ln(2 x
+
1)
+ ln(2 x
-
3)
=
2 ln(2 x
-
2)
(8) log
2
(2 x
-
6)
=
3
+ log
2 x
(9) log 5 x
2
8
-
3
= log
5 x
40
(10) log 2 x
+ log(4
-
16 x
)
=
2 log( x
-
2)
11. Fifty people were treated for a virus on the same day. The virus is highly contagious and the patients must stay in the hospital until the have no symptoms. The number of people p who show symptoms after t days can be modeled by the function: p
=
52.76
1
+
0.03
e
0.75
t
Show algebraically, how many days it will take for only one person to show symptoms.
Pre-Calculus Honors Homework: Complete any unfinished classwork