Approximating "e" and The Natural Logarithms (ln)

Homework: Exercise 25 #1-15
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Mr. Maks told us that the general idea of the lesson was that: loge=ln, in the case that e~2.71828...

We used the formulas: f(n) = 1+1/n , g(n)= (1+1/n)^n (for values n E I)
and Mr. Maks showed us, using Excel, the comparison between the two formulas using numbers between 1 and 1000.

We were also taught about Natural Exponential Functions and what they look like.
The graph of f(x) = e^x has these properties: (Important to know and understand)

• D= R (reals)
• R= (y>0)
• increases on its entire interval (moves upwards to the right)
• concave up
• one to one (** means it has an inverse**) ; if e^x1 = e^x2 (therefore, x1 = x2)
• 0<> 1 for x> 0

This relates back to the important equations Mr. Maks gave us yesterday of:
e^1nx=x and f(f^-1(x))=x (try them out on your calculator, they actually work!)

We also sketched some Exponential Graphs on Graphmatica
a)y= e^x-2
Compared to y=e^x, this graph is shifted 2 units down.

b)y=-e^-x
Compared to y=e^x, this graph is flipped over the y-axis, then the x- axis.

c)y= abs(e^x-1)
Compared to y=e^x, this graph is moved 1 unit down, then taking the absolute value of it causes the negative part of the graph to be bounced up and over onto the positive side of the x-axis.

If you compare graphs a, b, and c to the original equation (y=e^x) on Graphmatica, being able to see the actual changes really help. Also, if you know the approximate look of the original equation, you can just apply what we learned in our transformation unit and visualize where to move the graph by how much and so on. (It's pretty neat how all math builds on each other for the most part!)

Also, like Mr. Maks told us there's really great explinations and examples of logs in the Mickelson book starting on page 37.