Name: ____________________
Pre- Calculus 12
Date: _____________
Chapter 8 – Logarithmic Functions
Solving Logarithmic Equations work sheet
Solve each of the following equations for x:
1.
log(x) + log(x+9) = 1
2.
log(x) – log(x + 3) = 1
3.
log(x + 9) – log(x) = 1
4.
log(2x + 1) – log(x – 9) = 1
5.
log4(x + 3) + log4(x – 3) = 2
6.
log8(x + 1) – log8(x) = log8(4)
7.
log(x2 ) = (log x)2
8.
(log3x)2 – log3(x2) = 3
9.
log x = log x
10.
log(log x) = 2
11.
log5 x2 + 1 = 1
12.
log x2 + log x4 = log(2–3 )
13.
log6x + log6(x + 1) = 1
14.
log9(x + 1) = 2 + log9x
15.
log2(x + 4) = 2 – log2(x + 1)
16.
log(2x + 4) + log(x – 2) = 1
17.
ln x + ln(x + 1) = ln 2
18.
log(x + 3) – log(x – 2) = 2
19.
log2(2x2 + 4) = 5
20.
log(x – 6) + log(x + 3) = 1
21.
log(x) – log(5) = log(2) – log(x – 3)
22.
(ln x)3 = ln(x4)
23.
log(6x + 5) – log(3) = log(2) – log(x)
24.
(log x)3 = log(x4)
25.
log(x2) – log(x – 1) = 1
26.
log(x + 1) = log(2x2 + 3) – log(2x – 5) –1
27.
(log x)2 – 3 log(x) + 2 = 0
28.
1
3
29.
3 + log2(3) + log2(x) = log2(96)
30.
log(2x) – 2 log(x) = –1
31.
2

1 + x 
1 + x 
log2 1 – x  – 5 log2 1 – x  + 4 = 0





32.
[log(2x – 1)]2 – 3Log(2x – 1) – 10 = 0
33.
1
2
=
log(x) – 1
log(x) + 1
34.
log2(3 – x) + log2(1 – x) = 3
35.
Log3(2 – x) + Log3(4 – x) = 1
36.
Log5(2 – x) + Log5(6 – x) = 1
37.
Log2(3 – x) + Log2(7 – x) = 5
38.
Log3(5 – x) + Log3(3 – x) = 1
39.
Log2(3 – x) + Log2(6 – x) = 2
40.
Log2(7 – x) + Log2(5 – x) = 3
1
3
3
1
log(27) + log(9 – 3) = log(x)
Logarithms
41.
Log6(7 – x) + Log6(1 – x) = 3
42.
Log4(4 – x) + Log4(1 – x) = 1
43.
log3x + log9x + log27x = 5.5
44.
log(x – 3) + log(x + 6) = log(2) + log(5)
45.
log(x – 4) + log(x + 3) = log(5x + 4)
46.
ln(x2 +1) – 2 ln(x2 – 2x + 1) = ln(5)
47.
log5(x – 2) + 2 log5(x3 – 2) + log5(x – 2)–1 = 4
48.
2 log3(x – 2) + log3(x – 4)2 = 0
49.
log2(x + 2)2 + log2(x + 10)2 = 4 log2(3)
50.
x – 2
3x – 7
log2
 = log23x – 1
x
–
1




52.
log3(5x – 2) – 2 log3 3x + 1 = 1 – log3(4)
53.
log(3x – 2) – 2 = 2 log(x + 2) – log(50)
54.
1
51.
x – 7
x – 1
2 log2
 + log2x + 1 = 1
x
–
1




log(10x2)  log(x) = 1
55.
2 log9(x) + 9 logx(3) = 10
56.
logx(125x)  (log25x)2 = 1
57.
Log2x + 2Logx8 = 5
58.
Log3x + 2Logx9 = 5
59.
Log3x – Logx27 = 2
60.
Log4x + 4Logx64 = 8
61.
Log
62.
Log
63.
log(log x) + log(log x3 – 2) = 0
64.
log3x + 7(9 + 12x + 4x2) = 4 – log2x + 3(6x2 + 23x + 21)
65.
x2 logx(27)  log9(x) = x + 4
67.
log.5xx2 – 14 log16xx3 + 40 log4x x = 0
68.
70.
1
3
x + 4Logx27 = 14
2
x + 3Logx4 = 8
7
66.
logx(2) – log4(x) + 6 = 0
xlog(x) = 100x
69.
x3
xlog(x) = 100
(x + 1)log(x + 1) = 100(x + 1)
71.
xlog(x) = 3x
73.
Solve for t: P = P0ekt
75.
Solve for n: PVn = c
3
log3(x)
1
72.
2
= 64
74.
Solve for t: T = T0 + (T1 – T2)ekt
1
76.
Solve for Q: LogaQ = 3 Logay + b
77.
Solve for x: Logax = b + Logab
78.
Solve for y: Logay = Logab – a
79.
Solve for p: LogbP = b – Logb(aP)
2
Logarithms
Answers
3
1.
1
2.
no solution
3.
1
4.
118
5.
5
6.
1
3
7.
1; 100
8.
27; 3
9.
1; 104
1
1
10.
10100
11.
2 6
12.
15.
0
16.
3
2 2
13.
2
14.
1
2
17.
1
18.
299
19.
 14
20.
7
21.
5
22.
1; e2; e–2
23.
2
3
24.
1; 100; .01
25.
5
26.
5 + 131
6
27.
10; 100
28.
18
29.
4
30.
20
31.
1 15
;
3 17
32.
.0505; 2(105 + 1)
33.
103
34.
–1
35.
39.
43.
1
1
27
36.
40.
44.
1
3
4
37.
41.
45.
1
–11
8
38.
42.
46.
2
0
2; 3
47.
3
48.
3+ 2,3
49.
–1
50.
no solution
51.
–17
52.
1
53.
2
54.
1
;
10
55.
3; 39
56.
5; 625
1
57.
8, 4
58.
81, 3
59.
3, 27
60.
64, 4096
61.
8, 2
62.
729, 3
63.
10;
64.
–4
65.
2
66.
8;
15
1
3
67.
1; 4;
71.
.1223
10
1
2
68.
72.
5
1
1
100; 10
69.
10; 100
70.
10
1
3
4
9
99; – 10
1
3
3
Logarithms