natural logarithms(-notations-equations-and applications)

REMEMBER! loge = ln





OK, for this first slide he used 3 logarithmic identities to form the single logarithm
1. ln(x-1) + 3ln(x+3) becoming the numerator(x-1)(x+3)^3 from
the identity log e (MN)= log e M + log e N

2.from (ln(x-1)+3ln(x+3))-1/2ln(x^2+2) he got the denominator Square Root of (x^2+2) which is the same as(x^2+2)^1/2 from the identity log e (M/N) = log e M - log e N

3. and the third identity is log e (M^N) = N log e M which you can see done with both the numerator and denominator




the only thing i think i need to explain with this one is that if both sides of an equation have loge
in front we know we can take it away, correct? so i guess the opposite can be done as well where we add in loge, which is seen done in the second line (the red ink).


(remember! loge = ln)!



*lne=1


this slide uses the third identity mentioned in the first slide, and takes away ln from both sides(as mentioned in slide two), other than that i think it is self explanatory.




in this slide he adds ln to both slides making lne on the left side which previously stated equals one(see lime green * in above slides explination).

and again he used the third identity mentioned in slide 1.

Hope this helped you in anyway, shape, or for possible


Mr. Maksymchuk gave use another one of his shirt ideas this one was based on the Pink Floyd album cover the dark side of the moon with the rainbow and prisms, but his shirt would say Pink Freud. I didn't get the reference but maybe you did.


I later googled it and it was referanced to Sigmund Freud who was a phsycoanalysis who was best know for his theories of the unconcious mind and the defence mechanism of repression.


after all that i still didn't completely get it, oh well, my loss I guess

Approximating "e" and The Natural Logarithms (ln)

Homework: Exercise 25 #1-15
Also, remember to vote for which day you think our next test should be!

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.