Moisés Grillo Ing. Industrial VIDEOSDEMATEMATICAS.COM

𝐹𝑜𝑟𝑚𝑢𝑙𝑎 sin 𝑎𝑛𝑖𝑚𝑎𝑐𝑖𝑜𝑛 𝐴𝑛𝑖𝑚𝑎𝑑𝑎

Ecuaciones Logarítmicas con diferente base Profesor Moisés Grillo Derechos reservados Prohibida su reproducción

Cambio de base = 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 2 3 = 𝑙𝑜𝑔 2 𝑥 3 número = 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 8 = 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 2 2 3 = 𝑙𝑜𝑔 2 𝑥 3 = 1 3 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 8 𝑥 base = 1 3 𝑙𝑜𝑔 2 𝑥 𝑙𝑜𝑔 8 𝑥 = 𝑙𝑜𝑔 2 3 𝑥

𝑙𝑜𝑔 9 𝑥= 𝑙𝑜𝑔 3 2 𝑥 = 1 2 𝑙𝑜𝑔 3 𝑥 𝑙𝑜𝑔 100 𝑛= 𝑙𝑜𝑔 10 2 𝑛 = 1 2 𝑙𝑜𝑔 10 𝑛 𝑙𝑜𝑔 8 𝑦= 𝑙𝑜𝑔 2 3 𝑦 = 1 3 𝑙𝑜𝑔 2 𝑦 𝑙𝑜𝑔 32 3= 𝑙𝑜𝑔 2 5 3 = 1 5 𝑙𝑜𝑔 2 3 𝑙𝑜𝑔 16 𝑐= 𝑙𝑜𝑔 2 4 𝑐 = 1 4 𝑙𝑜𝑔 2 𝑐 𝑙𝑜𝑔 27 2= 𝑙𝑜𝑔 3 3 2 = 1 3 𝑙𝑜𝑔 3 2

𝑙𝑜𝑔 4 𝑥+ 𝑙𝑜𝑔 8 𝑥= 5 6 4 2 8 2 2 2 4 2 𝑙𝑜𝑔 2 2 𝑥+ 𝑙𝑜𝑔 2 3 𝑥= 5 6 1 2 2 1 1 2 𝑙𝑜𝑔 2 𝑥+ 1 3 𝑙𝑜𝑔 2 𝑥= 5 6 4= 2 2 8= 2 3

3+2 6 𝑙𝑜𝑔 2 𝑥= 5 6 𝑙𝑜𝑔 4 𝑥+ 𝑙𝑜𝑔 8 𝑥= 5 6 𝑙𝑜𝑔 2 2 𝑥+ 𝑙𝑜𝑔 2 3 𝑥= 5 6 5 6 𝑙𝑜𝑔 2 𝑥= 5 6 1 2 𝑙𝑜𝑔 2 𝑥+ 1 3 𝑙𝑜𝑔 2 𝑥= 5 6 𝑙𝑜𝑔 2 𝑥=1 𝑥= 2 1 1 2 + 1 3 𝑙𝑜𝑔 2 𝑥= 5 6 𝑥=2

6 𝑙𝑜𝑔 9 𝑥− 𝑙𝑜𝑔 3 𝑥=4 6𝑙𝑜𝑔 3 2 𝑥− 𝑙𝑜𝑔 3 1 𝑥=4 𝑙𝑜𝑔 3 𝑥= 4 2 6 2 𝑙𝑜𝑔 3 𝑥− 1 1 𝑙𝑜𝑔 3 𝑥=4 𝑙𝑜𝑔 3 𝑥=2 3 𝑙𝑜𝑔 3 𝑥−1 𝑙𝑜𝑔 3 𝑥=4 𝑥= 3 2 𝑥=9 2 𝑙𝑜𝑔 3 𝑥=4

𝑙𝑜𝑔 2 𝑥+2 𝑙𝑜𝑔 4 𝑥=6 8 𝑙𝑜𝑔 9 𝑥+2 𝑙𝑜𝑔 3 𝑥=6 𝑥=8 𝑥=3 3 𝑙𝑜𝑔 8 𝑥+5 𝑙𝑜𝑔 2 𝑥=12 2 𝑙𝑜𝑔 3 𝑥−3 𝑙𝑜𝑔 27 𝑥=2 𝑥=4 𝑥=9 10 𝑙𝑜𝑔 4 𝑥−6 𝑙𝑜𝑔 8 𝑥=3 4 𝑙𝑜𝑔 27 𝑥− 𝑙𝑜𝑔 9 𝑥= 5 2 𝑥=2 𝑥=27 3 𝑙𝑜𝑔 4 𝑥+2 𝑙𝑜𝑔 8 𝑥= 13 6 𝑥=2

2 𝑙𝑜𝑔 27 𝑥+ 𝑙𝑜𝑔 9 𝑥− 𝑙𝑜𝑔 81 𝑥= 11 12 2 𝑙𝑜𝑔 3 3 𝑥+ 𝑙𝑜𝑔 3 2 𝑥− 𝑙𝑜𝑔 3 4 𝑥= 11 12 2 3 𝑙𝑜𝑔 3 𝑥+ 1 2 𝑙𝑜𝑔 3 𝑥− 1 4 𝑙𝑜𝑔 3 𝑥= 11 12 2 3 + 1 2 − 1 4 𝑙𝑜𝑔 3 𝑥= 11 12

2 3 + 1 2 − 1 4 𝑙𝑜𝑔 3 𝑥= 11 12 3−2−4 2 3 1 2 2 8+6−3 12 𝑙𝑜𝑔 3 𝑥= 11 12 3 1 3 1 12 11 12 𝑙𝑜𝑔 3 𝑥= 11 12 𝑥=3 𝑙𝑜𝑔 3 𝑥=1

7 𝑙𝑜𝑔 25 𝑥− 2 3 𝑙𝑜𝑔 5 𝑥− 𝑙𝑜𝑔 125 𝑥=5 7 𝑙𝑜𝑔 5 2 𝑥− 2 3 𝑙𝑜𝑔 5 1 𝑥− 𝑙𝑜𝑔 5 3 𝑥=5 7 2 𝑙𝑜𝑔 5 𝑥− 2 3 𝑙𝑜𝑔 5 𝑥− 1 3 𝑙𝑜𝑔 5 𝑥=5 7 2 − 2 3 − 1 3 𝑙𝑜𝑔 5 𝑥=5

7 2 − 2 3 − 1 3 𝑙𝑜𝑔 5 𝑥=5 1 2 𝑙𝑜𝑔 5 𝑥=1 7 2 − 3 3 𝑙𝑜𝑔 5 𝑥=5 𝑙𝑜𝑔 5 𝑥=2 𝑥= 5 2 7 2 −1 𝑙𝑜𝑔 5 𝑥=5 𝑥=25 5 2 𝑙𝑜𝑔 5 𝑥=5

3𝑙𝑜𝑔 8 𝑥+2 𝑙𝑜𝑔 4 𝑥+5 𝑙𝑜𝑔 2 𝑥=21 𝑥=8 4 𝑙𝑜𝑔 27 𝑥+ 𝑙𝑜𝑔 3 𝑥−2 𝑙𝑜𝑔 81 𝑥= 11 3 𝑥=9 7 𝑙𝑜𝑔 5 𝑥−6 𝑙𝑜𝑔 25 𝑥−9 𝑙𝑜𝑔 125 𝑥=1 𝑥=5 𝑥=100 12 𝑙𝑜𝑔 1000 𝑥−2 𝑙𝑜𝑔 10 𝑥+6 𝑙𝑜𝑔 100 𝑥=10

Cambio de base = 𝑙𝑜𝑔 5 𝑥 𝑙𝑜𝑔 5 5 = 𝑙𝑜𝑔 5 𝑥 1 2 𝑙𝑜𝑔 5 5 número = 𝑙𝑜𝑔 5 𝑥 𝑙𝑜𝑔 5 5 = 𝑙𝑜𝑔 5 𝑥 1 2 𝑙𝑜𝑔 5 5 = 1 1 1 2 𝑙𝑜𝑔 5 𝑥 𝑙𝑜𝑔 5 𝑥 =2 𝑙𝑜𝑔 5 𝑥 base 𝑙𝑜𝑔 5 𝑥 = 𝑙𝑜𝑔 5 2 𝑥 2 = 𝑙𝑜𝑔 5 𝑥 2 =2 𝑙𝑜𝑔 5 𝑥

𝑙𝑜𝑔 6 𝑥= 𝑙𝑜𝑔 6 2 𝑥 2 =2 𝑙𝑜𝑔 6 𝑥 𝑙𝑜𝑔 3 7 𝑦= 𝑙𝑜𝑔 3 7 3 𝑦 3 =3 𝑙𝑜𝑔 7 𝑦 𝑙𝑜𝑔 5 2 𝑐= 𝑙𝑜𝑔 5 2 5 𝑐 5 =5 𝑙𝑜𝑔 2 𝑐 𝑙𝑜𝑔 7 3 𝑦= 𝑙𝑜𝑔 7 3 7 𝑦 7 =7 𝑙𝑜𝑔 3 𝑦 𝑙𝑜𝑔 6 9 𝑐= 𝑙𝑜𝑔 6 9 6 𝑐 6 =6 𝑙𝑜𝑔 9 𝑐

𝑙𝑜𝑔 5 𝑥+ 𝑙𝑜𝑔 3 5 𝑥=15 𝑙𝑜𝑔 5 2 𝑥 2 + 𝑙𝑜𝑔 3 5 3 𝑥 3 =15 𝑙𝑜𝑔 5 𝑥 2 + 𝑙𝑜𝑔 5 𝑥 3 =15 𝑙𝑜𝑔 5 𝑥=3 2 𝑙𝑜𝑔 5 𝑥+3 𝑙𝑜𝑔 5 𝑥=15 𝑥= 5 3 5 𝑙𝑜𝑔 5 𝑥=15 𝑥=125

𝑙𝑜𝑔 7 𝑥+ 𝑙𝑜𝑔 49 𝑥= 5 2 2+ 1 2 𝑙𝑜𝑔 7 𝑥= 5 2 𝑙𝑜𝑔 7 2 𝑥 2 + 𝑙𝑜𝑔 7 2 𝑥= 5 2 5 2 𝑙𝑜𝑔 7 𝑥= 5 2 𝑙𝑜𝑔 7 𝑥 2 + 1 2 𝑙𝑜𝑔 7 𝑥= 5 2 𝑙𝑜𝑔 7 𝑥=1 𝑥=7 2 𝑙𝑜𝑔 7 𝑥+ 1 2 𝑙𝑜𝑔 7 𝑥= 5 2

𝑙𝑜𝑔 2 𝑥+ 𝑙𝑜𝑔 3 2 𝑥=20 𝑙𝑜𝑔 8 𝑥+ 𝑙𝑜𝑔 3 2 𝑥= 20 3 𝑥=16 𝑥=4 𝑙𝑜𝑔 4 5 𝑥+ 𝑙𝑜𝑔 3 5 𝑥=7 𝑙𝑜𝑔 3 𝑥+ 𝑙𝑜𝑔 9 𝑥= 5 2 𝑥=5 𝑥=3 𝑙𝑜𝑔 3 𝑥+ 𝑙𝑜𝑔 5 3 𝑥=21 𝑥=27 𝑙𝑜𝑔 25 𝑥+ 𝑙𝑜𝑔 7 5 𝑥=30 𝑙𝑜𝑔 5 7 𝑥+ 𝑙𝑜𝑔 7 7 𝑥=12 𝑥=625 𝑥=7

Cambio de base 9 𝑙𝑜𝑔 8 6 = 𝑙𝑜𝑔 6 6 𝑙𝑜𝑔 6 𝑥 = 1 𝑙𝑜𝑔 6 𝑥 =9 𝑙𝑜𝑔 6 8 9 𝑙𝑜𝑔 8 6 = 𝑙𝑜𝑔 6 6 𝑙𝑜𝑔 6 𝑥 = 1 𝑙𝑜𝑔 6 𝑥 =9 𝑙𝑜𝑔 6 8 𝑙𝑜𝑔 𝑥 6 = 4 𝑙𝑜𝑔 3 2 4 𝑙𝑜𝑔 2 3 7 𝑙𝑜𝑔 3 5 =7 𝑙𝑜𝑔 5 3

8 𝑙𝑜𝑔 2 𝑥 = 1 𝑙𝑜𝑔 6 3 = 8 𝑙𝑜𝑔 𝑥 2 𝑙𝑜𝑔 3 6 9 𝑙𝑜𝑔 𝑥 4 = 17 𝑙𝑜𝑔 𝑦 𝑤 = 9 𝑙𝑜𝑔 4 𝑥 17 𝑙𝑜𝑔 𝑤 𝑦 12 𝑙𝑜𝑔 𝑏 𝑎 = 2 𝑙𝑜𝑔 𝑛 3 = 12 𝑙𝑜𝑔 𝑎 𝑏 2 𝑙𝑜𝑔 3 𝑛 5 𝑙𝑜𝑔 7 8 = 5 𝑙𝑜𝑔 8 7

7 𝑙𝑜𝑔 5 4 8 𝑙𝑜𝑔 𝑛 𝑥 7 𝑙𝑜𝑔 4 5= 8 𝑙𝑜𝑔 𝑥 𝑛= 6 𝑙𝑜𝑔 9 𝑥 3 𝑙𝑜𝑔 7 2 6 𝑙𝑜𝑔 𝑥 9= 3 𝑙𝑜𝑔 2 7= 4 𝑙𝑜𝑔 𝑐 𝑚 9 𝑙𝑜𝑔 7 𝑦 4 𝑙𝑜𝑔 𝑚 𝑐= 9 𝑙𝑜𝑔 𝑦 7=

5 𝑙𝑜𝑔 2 𝑥 + 2 𝑙𝑜𝑔 3 𝑥 =1 𝑙𝑜𝑔 𝑥 32×9 =1 288= 𝑥 1 5 𝑙𝑜𝑔 𝑥 2+2 𝑙𝑜𝑔 𝑥 3=1 𝑥=288 𝑙𝑜𝑔 𝑥 2 5 + 𝑙𝑜𝑔 𝑥 3 2 =1 𝑙𝑜𝑔 𝑥 32+ 𝑙𝑜𝑔 𝑥 9=1

2 𝑙𝑜𝑔 5 𝑥 + 1 3 𝑙𝑜𝑔 8 𝑥 =7 𝑙𝑜𝑔 𝑥 25×2 =7 50= 𝑥 7 2 𝑙𝑜𝑔 𝑥 5+ 1 3 𝑙𝑜𝑔 𝑥 8=7 𝑥 7 =50 𝑙𝑜𝑔 𝑥 5 2 + 𝑙𝑜𝑔 𝑥 3 8 =7 𝑥= 7 50 𝑙𝑜𝑔 𝑥 25+ 𝑙𝑜𝑔 𝑥 2=7

1 𝑙𝑜𝑔 7 𝑥 + 2 𝑙𝑜𝑔 2 𝑥 =1 𝑥=28 2 𝑙𝑜𝑔 2 𝑥 + 3 𝑙𝑜𝑔 3 𝑥 =1 𝑥=108 3 𝑙𝑜𝑔 2 𝑥 + 1 2 𝑙𝑜𝑔 9 𝑥 =3 𝑥= 3 24 1 5 𝑙𝑜𝑔 32 𝑥 + 1 3 𝑙𝑜𝑔 8 𝑥 =5 𝑥= 5 4

𝑙𝑜𝑔 5 𝑥+3 𝑙𝑜𝑔 3 𝑥=2 𝑙𝑜𝑔𝑥= 2 7,718386381 𝑙𝑜𝑔𝑥 𝑙𝑜𝑔5 +3 𝑙𝑜𝑔𝑥 𝑙𝑜𝑔3 =2 𝑙𝑜𝑔𝑥=0,2591215186 10 𝑥= 10 0,2591215186 1 𝑙𝑜𝑔5 + 3 𝑙𝑜𝑔3 𝑙𝑜𝑔𝑥=2 𝑥=1,816 7,718386381𝑙𝑜𝑔𝑥=2

5 7 𝑙𝑜𝑔 6 𝑥+ 4 3 𝑙𝑜𝑔 2 𝑥=11 𝑙𝑜𝑔𝑥= 11 5,347164038 5𝑙𝑜𝑔𝑥 7𝑙𝑜𝑔6 + 4𝑙𝑜𝑔𝑥 3𝑙𝑜𝑔2 =11 𝑙𝑜𝑔𝑥=2,057165242 10 𝑥= 10 2,057165242 5 7𝑙𝑜𝑔6 + 4 3𝑙𝑜𝑔2 𝑙𝑜𝑔𝑥=11 𝑥=114,1 5,347164038𝑙𝑜𝑔𝑥=11

6 𝑙𝑜𝑔 5 𝑥+7 𝑙𝑜𝑔 2 𝑥=6 5 7 𝑙𝑜𝑔 3 𝑥+ 4 3 𝑙𝑜𝑔 4 𝑥=1 𝑥=1,543 𝑥=1,86 4 𝑙𝑜𝑔 3 𝑥+5 𝑙𝑜𝑔 8 𝑥=10 6 5 𝑙𝑜𝑔 6 𝑥+ 8 9 𝑙𝑜𝑔 9 𝑥=3 𝑥=5,229 𝑥=16,32 3 𝑙𝑜𝑔 12 𝑥+2 𝑙𝑜𝑔 15 𝑥=12 𝑥=476,8 4 9 𝑙𝑜𝑔 2 𝑥+ 6 7 𝑙𝑜𝑔 5 𝑥=2 2 3 𝑙𝑜𝑔 7 𝑥+ 4 5 𝑙𝑜𝑔 8 𝑥=5 𝑥=5,5 𝑥=967,4

2 3 𝑙𝑜𝑔 7 𝑥+ 4 5 𝑙𝑜𝑔 8 𝑥=5 𝑙𝑜𝑔𝑥=2,98559 𝑥= 10 2,98559 2𝑙𝑜𝑔𝑥 3𝑙𝑜𝑔7 + 4𝑙𝑜𝑔𝑥 5𝑙𝑜𝑔8 =5 𝑥=967,36 2 3𝑙𝑜𝑔7 + 4 5𝑙𝑜𝑔8 𝑙𝑜𝑔𝑥=5 𝑙𝑜𝑔𝑥= 5 2 3𝑙𝑜𝑔7 + 4 5𝑙𝑜𝑔8

𝑙𝑜𝑔 16 𝑐= 𝑙𝑜𝑔 2 4 𝑐 = 1 4 𝑙𝑜𝑔 2 𝑐 1 2 𝑙𝑜𝑔 4 𝑐 = 1 2 𝑙𝑜𝑔 2 2 𝑐 𝑙𝑜𝑔 16 𝑐= 𝑙𝑜𝑔 4 2 𝑐 = 1 2 𝑙𝑜𝑔 4 𝑐 = 1 2×2 𝑙𝑜𝑔 2 𝑐 = 1 4 𝑙𝑜𝑔 2 𝑐