Funciones racionales, irracionales, exponenciales y logaritmicas
Ecuaciones logaritmicas
Transcript of Ecuaciones logaritmicas
![Page 1: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/1.jpg)
ECUACIONES LOGARITMICAS
Profesora Srta. Yanira Castro Lizana
![Page 2: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/2.jpg)
![Page 3: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/3.jpg)
Ecuaciones Logarítmicas
Caso I
![Page 4: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/4.jpg)
Ecuaciones Logarítmicas (Caso I)
log3 (2 x - 1) = 2
![Page 5: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/5.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1log3 (2 x - 1) = 2
![Page 6: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/6.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1
9 = 2 x - 1
log3 (2 x - 1) = 2
![Page 7: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/7.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1
9 = 2 x - 1
9 + 1 = 2 x
log3 (2 x - 1) = 2
![Page 8: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/8.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1
9 = 2 x - 1
9 + 1 = 2 x
10 = 2 x
log3 (2 x - 1) = 2
![Page 9: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/9.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1
9 = 2 x - 1
9 + 1 = 2 x
10 = 2 x
= x10
2
log3 (2 x - 1) = 2
![Page 10: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/10.jpg)
Ecuaciones Logarítmicas (Caso I)
32 = 2 x - 1
9 = 2 x - 1
9 + 1 = 2 x
10 = 2 x
= x10
2
log3 (2 x - 1) = 2
![Page 11: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/11.jpg)
Ecuaciones Logarítmicas
Caso II
![Page 12: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/12.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
![Page 13: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/13.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
![Page 14: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/14.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1
![Page 15: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/15.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
![Page 16: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/16.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
![Page 17: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/17.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
10 x – 30 = x + 6
![Page 18: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/18.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
10 x – 30 = x + 6
10 x – x = 30 + 6
![Page 19: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/19.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
10 x – 30 = x + 6
10 x – x = 30 + 6
9 x = 36
![Page 20: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/20.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
10 x – 30 = x + 6
10 x – x = 30 + 6
9 x = 36
36x
9
![Page 21: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/21.jpg)
Ecuaciones Logarítmicas (Caso II)
log (x + 6) = 1 + log (x – 3)
log (x + 6) - log (x – 3) = 1
logx 6x 3
= 1 101 =
x 6x 3
10 (x – 3) = x + 6
10 x – 30 = x + 6
10 x – x = 30 + 6
9 x = 36
36x
9
![Page 22: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/22.jpg)
Ecuaciones Logarítmicas
Caso II (Continuación)
![Page 23: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/23.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
![Page 24: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/24.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
![Page 25: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/25.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
![Page 26: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/26.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
![Page 27: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/27.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
![Page 28: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/28.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
16 83 3e x
![Page 29: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/29.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
16 83 3e x
33 16 8e x
![Page 30: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/30.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
16 83 3e x
33 16 8e x
16 8e x
![Page 31: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/31.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
16 83 3e x
33 16 8e x
16 8e x8 16e x2
![Page 32: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/32.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
2
3
x 16ln lnx
3x
2
3
x.x 16ln
3x
3
13
x 16ln
3x
83 16
lnx3
16 83 3e x
33 16 8e x
16 8e x8 16e x2
![Page 33: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/33.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
![Page 34: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/34.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
![Page 35: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/35.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
![Page 36: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/36.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
161lnx. 1 23 3
![Page 37: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/37.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
161lnx. 1 23 3
168lnx. 3 3
![Page 38: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/38.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
161lnx. 1 23 3
168lnx. 3 3
16 3lnx .
3 8
2
![Page 39: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/39.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
161lnx. 1 23 3
168lnx. 3 3
16 3lnx .
3 8
lnx 2
2
![Page 40: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/40.jpg)
Ecuaciones Logarítmicas (Caso II Continuación)
23 16lnx ln x lnx
3
123 16
lnx lnx lnx3
1 16lnx .lnx 2.lnx
3 3
161lnx. 1 23 3
168lnx. 3 3
16 3lnx .
3 8
lnx 2
2
![Page 41: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/41.jpg)
Ecuaciones Logarítmicas
Caso III
![Page 42: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/42.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 823log
![Page 43: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/43.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
23log
![Page 44: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/44.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
23log
![Page 45: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/45.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
23log
![Page 46: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/46.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
23log
![Page 47: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/47.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
23log
![Page 48: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/48.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
23log
![Page 49: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/49.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
23log
![Page 50: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/50.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1
23log
![Page 51: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/51.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1
9 – 1 = x1
23log
![Page 52: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/52.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1
9 – 1 = x1
23log
![Page 53: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/53.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1 3 – 4 = x2 + 1
9 – 1 = x1
23log
![Page 54: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/54.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1 3 – 4 = x2 + 1
9 – 1 = x1– 1 = x2
181
23log
![Page 55: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/55.jpg)
Ecuaciones Logarítmicas (Caso III)
(x + 1) + log3 (x + 1)2 = 8
| z = log3 (x + 1) |
z2 + 2 z = 8
z2 + 2 z – 8 = 0
z1 = 2 z2 = – 4
[log3 (x + 1)]2 + 2 log3 (x + 1) = 8
z1 = log3 (x1 + 1) z2 = log3 (x2 + 1)
2 = log3 (x1 + 1) – 4 = log3 (x2 + 1)
32 = x1 + 1 3 – 4 = x2 + 1
9 – 1 = x1– 1 = x2
181
23log
![Page 56: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/56.jpg)
Resolver:
![Page 57: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/57.jpg)
ECUACIONES EXPONENCIALES QUE SE RESUELVEN CON LOGARITMOS
Aquellas ecuaciones exponenciales que no se pueda expresar en términos de bases iguales, se utilizan los logaritmos y sus propiedades para hallar la solución.
![Page 58: Ecuaciones logaritmicas](https://reader035.fdocumento.com/reader035/viewer/2022062220/559039d81a28ab090d8b469b/html5/thumbnails/58.jpg)