Additional exercises
1. Solve the following equation:
2
1
27 · 81−x2
=
218
36
Solution:
x = ±2
2. Solve the following equation:
√
34
x
√
+5·9
x
= 126
Solution:
x=1
3. Solve the following equation:
3 · 16x + 2 · 81x = 5 · 36x
Solution:
x = 0, 21
4. Solve the following equation:
2x
2
−1
2
− 3x = 3x
2
−1
2
− 2x
+2
Solution:
√
x=± 3
5. Solve the following equation:
−2 · 3x
2
−7x+11
2
+ 32x
Solution:
x = 5, 2
1
−14x+21
= −3
6. Solve the following system of equations:
(
2x + y −
x + 2y −
1
x2
1
y2
=0
=0
3
3
2
2
Hint: use the formula:
x − y = (x − y) · x + xy + y
1
1
Solution: 3− 3 , 3− 3
7. Solve the following equation:
3
x(log x)
−5 log x
= 0.0001
Solution:
x = 100, 10, 0.1, 0.01
8. Solve the following equation:
x2
5
·
1
5x
2
=
1
√
5
x
Solution:
x = 0, 1.5
9. Solve the following equation:
log2 (2x − 1) − log2 3 = 2 − x
Solution:
x=2
10. Solve the following equation:
log3 (12 + 5 log2 (8 + log7 x)) = 3
Solution:
x=1
11. Solve the following equation:
log (log x) + log log x3 − 2 = 0
Solution:
x = 10
2
12. Solve the following equation:
2
2
log6 3x + 1 − log6 32−x + 9 = log6 2 − 1
Solution:
x = ±1
13. Solve the following equation:
log2 (4 · 3x − 6) − log2 (9x − 6) = 1
Solution:
x=1
14. Solve the following equation:
2
1
x1− 3 log x − √
3
1
=0
100
Solution:
x = 100, √110
15. Solve the following equation:
x
log x+5
3
= 105+log x
Solution:
x = 1000, 1015
16. Solve the following equation:
0.1 · x
1+log x
2−log x3
1
=
x
Solution:√
x = 10, 10
17. Solve the following equation:
5log x + xlog 5 = 50
Solution:
x = 100
3
18. Solve the following equation:
10 · x2 log
x3
2
x
=
x3 log x
10
ׁHint: use the formula: a3 + b3 = (a + b) · a2 − ab + b2
Solution:√
x=
1
10 ,
10, 100
19. Solve the following equation:
27log x − 7 · 9log x − 21 · 3log x + 27 = 0
Hint: use the formula: a3 + b3 = (a + b) · a2 − ab + b2
Solution:
x = 1, 100
20. Solve the following equation:
log9 16
log2 3 + 2 log4 x = x log3 x
Solution:
x=
16
3
21. The sum of the rst 8 terms in an arithmetic sequence equals to the 29th
term. How many consecutive terms must we add, starting from the rst
term, to get a sum that is equal to the 46th term?
Solution:
10 terms.
22. Given an arithmetic sequence where the sum of the rst 40 terms equals
to the sum of the rst 20 terms.
(a) Prove that 2a1 = −59d.
(b) Prove that the sum of the rst 60 terms equals to zero.
(c) It is given that the sequence is increasing (d is positive). Find the
location (index) of the rst positive term.
Solution:
31
4