6.6 Solving Exponential and
Logarithmic Equations
Solutions
Solve.
1.
log5 (5x + 10) = 2
2.
log3 (2x + 1) = 3
4.
log (10x + 20) = 3
52 = 5x + 10
25 = 5x + 10
–10 –10
15 = 5x
15 5x
=
5
5
3=x
Check: For x = 3, 5x + 10 = 15+10 = 25
Solution set:
3.
{3}
log (3x – 2) = 2
log10 (3x – 2) = 2
102 = 3x – 2
100 = 3x – 2
+2
+2
102 = 3x
102 3x
=
3
3
34 = x
Check: For x = 34: 3x – 2 = 100
Solution set:
{34}
Answers: 1. {3};
3. {34}
Math 102 6.6 Page 1 of 5 Solve.
5.
log10 (w2) = 2
6.
log5 (y2 – 2y + 1) = 2
102 = w2
100 = w2
± 100 = w 2
±10 = w
Argument Check: For w = 10, w2 = 100 <
for w = –10, w2 = 100 <
Solution set:
{10, –10}
Solve. Round your answers to 4 decimal places (x.xxxx)
7.
4x = 9
8.
3x = 5
log 4x = log 9
x log 4 = log 9
x log 4 log 9
=
log 4 log 4
x ≈ 1.584962501
x≈
Answers:
1.5850
5. {± 10}; 7. {≈1.5850}
Math 102 6.6 Page 2 of 5 Solve. Round your answers to 4 decimal places (x.xxxx)
9.
3x – 1 = 2
10.
22x + 1 = 5
12.
e 2x + 1 = 0.04
log 3x – 1 = log 2
(x – 1) log 3 = log 2
(x − 1)log 3 log 2
=
log 3
log 3
log 2
log 3
+1 +1
x −1 =
x=
log 2
+1
log 3
x ≈ 1.630929754
x≈
11.
1.6309
ex = 5
ln ex = ln 5
x ln e = ln 5
x • 1= ln 5
x = ln 5
x ≈ 1.609437912
x≈
Answers: 9. {≈1.6309};
Math 102 1.6094
11. {≈1.6094}
6.6 Page 3 of 5 Solve.
13.
log2 x = log2 7
14.
log4 (x + 2) = log4 8
16.
log3 (x2 – 5) = log3 (4x)
x=7
Argument Check: For x = 7, x = 7
<
Solution set: {7}
15.
log7 (x2 – 2) = log7 (x)
x2 – 2 = x
–x
–x
x2 – x – 2 = 0
–2
–1 2
1 –2
Sum –1
Product
(x + 1)(x – 2) = 0
x + 1 = 0 or x – 2 = 0
–1 –1
+2 +2
x = –1 or
x=2
Argument Check: For x = –1, x2 – 2 = –1 =
For x = 2, x2 – 2 = 4 – 2 = 2 <
Solution set: {2}
Answers:
13. {7};
Math 102 15.
{2}
6.6 Page 4 of 5 Solve.
17. log7 (x – 3) = log7 (6 – 2x) – log7 (3x) 18. log6 (x – 5) + log6 (x + 1) = log6 (x + 19)
6 − 2x
log7 (x – 3) = log 7
3x
6 − 2x
x–3 =
3x
6 − 2x
3x • (x – 3) =
• 3x
3x
3x2 – 9x = 6 – 2x
+2x
+2x
Product –18
2
3x – 7x = 6
–1
18
–6 –6
–2
9
3x2 – 7x – 6 = 0
2
–9
3x2 + 2x – 9x – 6 = 0
Sum –7
(x – 3)(3x + 2) = 0
3x
2
x – 3 = 0 or 3x + 2 = 0
x
+3 +3
–2 –2
–3
x = 3 or
3x = –2
3x −2
=
3
3
2
x=−
3
3x2
– 9x
2x
–6
Argument Check: For x = 3, x – 3 = 0 =
2
2
8
For x = − , x − 3 = − − 3 = − =
3
3
3
Solution set:
Answers:
17.
Math 102 ∅
∅
6.6 Page 5 of 5