EXPONENCIÁLNÍ A LOGARITMICKÉ ROVNICE
(Semináˇr z matematiky I - M1130/02 2016)
(1) ˇRešte v R exponenciální rovnice:
(a) 23x−1 ·4 = 8x+1 ·
1
2
x
[2]
(b)
1
3x
= 1√
3
·
6
√
273−3x ·
1
9
x+3
[−2]
(c) 22x ·5x −22x−1 ·5x+1 = −600 [2]
(d) 2·0,5x2+8
3 x
=
8
3
√
4
−2;−
2
3
(e) 7·4−x+2 = 3·4−x+3 −5 [2]
(f) 9x−0,5 +90,5−x =
10
3
[0;1]
(g) 2x−1 −2x−2 = 5x−3 +2x−3 [3]
(h) 4x +6x = 2·9x [0]
(i) 2·4x +5x−1
2 = 5x+1
2 −22x−1 3
2
(j) 4+
√
15
x
+ 4−
√
15
x
= 8 [−2;2]
(k) 23x +23x−1 +23x−2 +... =
√
12·23x −8 0; 1
3
(l) 2x +2x−1 +2x−2 +... = 2
√
3·2x +4 [2]
(2) ˇRešte v R logaritmické rovnice:
(a) log(x−9)+2·log
√
2x−1 = 2 [13]
(b) log
√
3x−5+log
√
7x−3 = 1+log
√
0,11 [2]
(c)
log7x
log(2x−7)
= 2 [7]
(d) log x2 log
√
x−log
1
x
= 2 [10;0,01]
(e) logloglogx = 0 1010
(f) log5 (2x+9)+log5 (4−3x) = 2+log5 (4+x) [−2]
(g) (log4 x−2)·log4 x =
3
2
·(log4 x−1) [2;64]
(h) log2+log 4x−2 +9 = 1+log 2x−2 +1 [2;4]
(i) log3 3x2−13x+28 +
2
9
= log5 0,2 [3;10]
(j)
√
xlog
√
x = 10
1
100
;100
(k) x
3
8 log3 x−3
4 logx
= 1000 [0,01;100]
(l) log3 x−2log1
3
x = 6 [9]
(m) logx−1 3 = 2 1+
√
3
(n) logx 9x2 ·log2
3 x = 4
1
9
;3
(o) log8 x+log2
8 x+log3
8 x+... =
1
2
[2]
(p) x
logx+5
3 = 105+logx 10−5;103