Logaritmos con cambio de base Profesor Moisés Grillo Ing. Industrial Derechos reservados Prohibida su reproducción

𝑙𝑜𝑔 32 8

= 3 5 𝑙𝑜𝑔 32 8 =𝑙𝑜𝑔 2 5 2 3 8 2 32 2 4 2 16 2 8 2 2 2 4 2 1 2 2 1

= 4 3 𝑙𝑜𝑔 27 81 =𝑙𝑜𝑔 3 3 3 4 81 3 27 3 27 3 9 3 3 3 9 3 1 3 3 1

= 2 3 = 1 5 𝑙𝑜𝑔 8 4 = 𝑙𝑜𝑔 2 3 2 2 𝑙𝑜𝑔 32 2 = 𝑙𝑜𝑔 2 5 2 3 = 6 4 = 3 2 = 2 3 𝑙𝑜𝑔 16 64 = 𝑙𝑜𝑔 2 4 2 6 𝑙𝑜𝑔 125 25 = 𝑙𝑜𝑔 5 3 5 2 2 = 1 4 = 1 3 𝑙𝑜𝑔 81 3 = 𝑙𝑜𝑔 3 4 3 1 𝑙𝑜𝑔 27 3 = 𝑙𝑜𝑔 3 3 3 1 = 1 2 = 3 5 𝑙𝑜𝑔 32 8 𝑙𝑜𝑔 49 7 = 𝑙𝑜𝑔 7 2 7 1 = 𝑙𝑜𝑔 2 5 2 3

𝑙𝑜𝑔 𝑛 𝑎 𝑏 = 𝑙𝑜𝑔 𝑛 𝑎− 𝑙𝑜𝑔 𝑛 𝑏 = 𝑙𝑜𝑔 2 1 2 2 𝑙𝑜𝑔 2 3 2 𝑙𝑜𝑔 3 2 1 4 = 𝑙𝑜𝑔 2 1 2 2 𝑙𝑜𝑔 2 3 2 𝑙𝑜𝑔 3 2 1 4 = 𝑙𝑜𝑔 3 2 1 2 2 Cambio de Base 𝑙𝑜𝑔 𝑥 1=0 𝑙𝑜𝑔 𝑎 𝑎 𝑛 =𝑛 𝑙𝑜𝑔 2 1 2 2 𝑙𝑜𝑔 2 3 2 = 𝑙𝑜𝑔 2 1− 𝑙𝑜𝑔 2 2 2 1 3 𝑙𝑜𝑔 2 2 = −2 1 3 1 =− 6 1 =−6 𝑙𝑜𝑔 𝑏 𝑛 𝑎 = 1 𝑛 𝑙𝑜𝑔 𝑏 𝑎 𝑙𝑜𝑔 𝑛 𝑎 𝑏 = 𝑙𝑜𝑔 𝑛 𝑎− 𝑙𝑜𝑔 𝑛 𝑏

𝑙𝑜𝑔 𝑛 𝑎 𝑏 = 𝑙𝑜𝑔 𝑛 𝑎− 𝑙𝑜𝑔 𝑛 𝑏 = 𝑙𝑜𝑔 3 5 3 𝑙𝑜𝑔 3 1 3 4 𝑙𝑜𝑔 1 81 5 3 = 𝑙𝑜𝑔 3 5 3 𝑙𝑜𝑔 3 1 3 4 𝑙𝑜𝑔 1 81 5 3 = 𝑙𝑜𝑔 1 3 4 5 3 𝑙𝑜𝑔 𝑏 𝑛 𝑎 = 1 𝑛 𝑙𝑜𝑔 𝑏 𝑎 Cambio de Base = 1 5 𝑙𝑜𝑔 3 3 𝑙𝑜𝑔 3 1− 𝑙𝑜𝑔 3 3 4 = 1 5 −4 𝑙𝑜𝑔 3 5 3 𝑙𝑜𝑔 3 1 3 4 =− 1 20 1 𝑙𝑜𝑔 𝑎 𝑎 𝑛 =𝑛 𝑙𝑜𝑔 𝑥 1=0 𝑙𝑜𝑔 𝑛 𝑎 𝑏 = 𝑙𝑜𝑔 𝑛 𝑎− 𝑙𝑜𝑔 𝑛 𝑏

= 𝑙𝑜𝑔 2 1 2 3 𝑙𝑜𝑔 2 3 2 2 𝑙𝑜𝑔 3 4 1 8 = 𝑙𝑜𝑔 3 2 2 1 2 3 = − 𝑙𝑜𝑔 2 2 3 2 3 𝑙𝑜𝑔 2 2 =− 9 2 = 𝑙𝑜𝑔 5 5 𝑙𝑜𝑔 5 1 5 = 1 2 𝑙𝑜𝑔 5 5 𝑙𝑜𝑔 5 1− 𝑙𝑜𝑔 5 5 = 1 2 0−1 =− 1 2 𝑙𝑜𝑔 1 5 5 = 𝑙𝑜𝑔 3 1 3 2 𝑙𝑜𝑔 3 4 3 𝑙𝑜𝑔 4 3 1 9 = 𝑙𝑜𝑔 4 3 1 3 2 = − 𝑙𝑜𝑔 3 3 2 1 4 𝑙𝑜𝑔 3 3 =−8 1 = 𝑙𝑜𝑔 2 5 2 2 𝑙𝑜𝑔 2 1 2 4 = 2 5 𝑙𝑜𝑔 2 2 − 𝑙𝑜𝑔 2 2 4 = 2 5 −4 𝑙𝑜𝑔 1 16 5 4 = 𝑙𝑜𝑔 1 2 4 5 2 2 =− 1 10 1 2

𝑙𝑜𝑔 2 2 4 8 3 2 = 𝑙𝑜𝑔 2 2 2 2 2 3 3 2 Cambio de Base = 𝑙𝑜𝑔 2 2 2 2 3 3 2 𝑙𝑜𝑔 2 2 2

𝑙𝑜𝑔 2 2 2 𝑙𝑜𝑔 2 2 2 2 3 3 2 1 2 𝑙𝑜𝑔 2 2 2 𝑙𝑜𝑔 2 2 2 2 3 − 𝑙𝑜𝑔 2 3 2 1 2 𝑙𝑜𝑔 2 2+ 𝑙𝑜𝑔 2 2 𝑙𝑜𝑔 2 2 2 + 𝑙𝑜𝑔 2 2 3 − 1 3 𝑙𝑜𝑔 2 2 1 2 1+ 1 2 𝑙𝑜𝑔 2 2 2+ 3 2 𝑙𝑜𝑔 2 2− 1 3 1 2 + 1 4 = 2+1 4 = 3 4 2+ 3 2 − 1 3 = 12+9−2 6 = 19 6

𝑙𝑜𝑔 2 2 4 8 3 2 = 𝑙𝑜𝑔 2 2 2 2 2 3 3 2 = 𝑙𝑜𝑔 2 2 2 2 3 3 2 𝑙𝑜𝑔 2 2 2 = 19 6 3 4 = 38 9 3 2

𝑙𝑜𝑔 5 7 3 7 7 3 7 7 Cambio de Base = 𝑙𝑜𝑔 7 3 7 7 𝑙𝑜𝑔 7 5 7 3 7 7 = 𝑙𝑜𝑔 7 3 7 7 𝑙𝑜𝑔 7 5 7 3 7 7 𝑙𝑜𝑔 5 7 3 7 7 3 7 7 = 1 3 𝑙𝑜𝑔 7 7 7 1 5 𝑙𝑜𝑔 7 7 3 7 7 = 5 𝑙𝑜𝑔 7 7 7 3 𝑙𝑜𝑔 7 7 3 7 7 Cambio de Base = 5 𝑙𝑜𝑔 7 7+ 𝑙𝑜𝑔 7 7 3 𝑙𝑜𝑔 7 7+ 𝑙𝑜𝑔 7 3 7 − 𝑙𝑜𝑔 7 7 = 5 1+ 1 2 3 1+ 1 3 − 1 2 = 5 2+1 2 3 6+2−3 6 = 5 3 2 3 5 6 = 15 2 5 2 = 15 5 =3 1 2

𝑙𝑜𝑔 3 3 3 27∙ 3 5 9 𝑙𝑜𝑔 8 32 4 4∙ 2 3 16 = 62 45 = 7 66 𝑙𝑜𝑔 5 5 5 5 4 5 5 𝑙𝑜𝑔 81 7 27 3 3 9 5 3 = 28 75 = 15 14 𝑙𝑜𝑔 5 6 6 7 6∙ 3 6 6 𝑙𝑜𝑔 3 10 4 10 6 10∙ 9 10 10 = 25 63 = 11 45