Leyes de los logaritmos

Logaritmos Naturales

Evaluació de expresiones con logaritmos

Graficación de funciones logarítmicas

Caracteristicas de la grafica de la funcion logaritmicas


Para construir una tabla de valores, se eligieron los valores para x como potencias de 2 de modo que pueda hallar con facilidad sus logaritmos.

Reflexion de graficas de funciones logaritmicas
Desplazamiento Vertical: loga X +_ c
Desplazamiento horizontal: loga (x+_k)

Logaritmos Comunes
El logaritmo con base 10 se llama logaritmo comun y se denota omitiendo la base
log X = log10 X

Evalue cada expresion con logaritmos
Para determinar con exactitud el valor de un logaritmo escribimos el logaritmo en notacion exponencial

Funcion Logaritmica

Modelo de Crecimiento

Ejemplo:

La cantidad inicial de bacterias en un cultivo es de 500 bacterias. Posteriormente un biólogo hace un conteo de muestra y encuentra que la tasa relativa de crecimiento es de 40% por hora. Encuentre la cantidad de bacterias a las 5,10 y 20 horas.

Función Exponencial Natural

Leonhard Euler

Leonhard Euler (1707-1783) was arguably the greatest mathematician of the eighteenth century (His closest competitor for that title is Lagrange) and one of the most prolific of all time; his publication list of 886 papers and books may be exceeded only by Paul Erdös. Euler's complete works fill about 90 volumes. Remarkably, much of this output dates from the the last two decades of his life, when he was totally blind.
Euler's important contributions were so numerous that terms like "Euler's formula" or "Euler's theorem" can mean many different things depending on context. Just in mechanics, one has Euler angles (to specify the orientation of a rigid body), Euler's theorem (that every rotation has an axis), Euler's equations for motion of fluids, and the Euler-Lagrange equation (that comes from calculus of variations). The "Euler's formula" with which most American calculus students are familiar defines the exponentials of imaginary numbers in terms of trigonometric functions. But there is another "Euler's formula" that (to use the modern terminology adopted long after Euler's death) gives the values of the Riemann zeta function at positive even integers in terms of Bernoulli numbers. There are both Euler numbers and Eulerian numbers, and they aren't the same thing. Euler's study of the bridges of Königsberg can be seen as the beginning of combinatorial topology (which is why the Euler characteristic bears his name).
Though born and educated in Basel, Switzerland, Euler spent most of his career in St. Petersburg and Berlin. He joined the St. Petersburg Academy of Sciences in 1727. In 1741 he went to Berlin at the invitation of Frederick the Great, but he and Frederick never got on well and in 1766 he returned to St. Petersburg, where he remained until his death. Euler's prolific output caused a tremendous problem of backlog: the St. Petersburg Academy continued publishing his work posthumously for more than 30 years. Euler married twice and had 13 children, though all but five of them died young.
Euler's powers of memory and concentration were legendary. He could recite the entire Aeneid word-for-word. He was not troubled by interruptions or distractions; in fact, he did much of his work with his young children playing at his feet. He was able to do prodigious calculations in his head, a necessity after he went blind. The contemporary French mathematician Condorcet tells the story of two of Euler's students who had independently summed seventeen terms of a complicated infinite series, only to disagree in the fiftieth decimal place; Euler settled the dispute by recomputing the sum in his head.

Funciones exponenciales y logarítmicas

Funciones exponenciales:
Esquema del capitulo

Se estudia una nueva forma de funciones llamadas funciones exponenciales
Las funciones exponenciales son apropiadas para modelar el crecimiento poblacional para los seres vivos