Solving Exponential Equations + Looking at exponential graphs.

I would like to start this off by saying sorry for the late blog post. Oh and that ..
ROXANNE WILL BE NEXT SCRIBE.
K thanks.

We started off our lesson with Mr. K's great line, "You are always looking for a pattern .."
SLIDE 3/10
Then the two questions. The one that people may have had problems with is the question with the solution in green.
Basically the same steps to solving exponential equations, but maybe a little more thinking.

49 becomes 7 ^ 2 .. All to the power of (x-2)
-You would then multiply 2 to (x-2)
.. 2(x-2)
.. 2x - 4
So the equation is now:
7 ^ 2x-4 = 7(7^(1/2))

I'm sure the square root caught most off guard. Both numbers have the same base so you add the exponents .. Which results in:
7 ^ 2x-4 = 7^(3/2)
..2x-4=3/2

Fractions are a nuisance at times, so we can multiply every term by two to get rid of it.
..4x-8=3

Thus solving it you'll find that x = 11/4

** YOU MAY ONLY MULTIPLY BASE NUMBERS IF EXPONENTS ARE SAME.
** YOU MAY ONLY ADD EXPONENTS OF BASE NUMBERS ARE THE SAME.

SLIDE 4/10

I was only caught off guard by the last question on this slide. It was a tricky one!
Again, you CANNOT multiply the base numbers unless the exponents are the same!
You had to take an extra step, and because all of us are smarter than the average bear, someone did.
The equation was divided by 2 to get rid of the useless 2.
..250/2 = (2(5))^2x-1/2
Now you have:
125 = 5^2x-1

Now it's the equation has become simply easy to solve.
-Work towards having the same base number, then solving for X.

5^3 = 5^2x-1

3 = 2x - 1
4 = 2x
2 = x

SLIDE 5/10

"Always looking for a pattern .."

May be tricky, new to the eyes, but definitely not new to the brain.

The equation is in QUADRATIC FORM. Meaning you must factor, but how? Where? When? .... Why?
There's another way to look at this equation. 3^2x is also ..
(3^x)^2
Making the equation become:
(3^x)^2 - 6(3^x) + 9 = 0

Ahhh... Now you see!

It would be easier, and you can do this. Let (3^x) = a .. or b .. or z .. or George
a^2 - 3a -3a + 9 = 0
(a - 3)(a - 3) = 0
Therefore, a = 3
Because we let (3^x) = a
.. (3^x) = 3^1
..X = 1

The second question is the same! Of course everyone got confused and did not accept REJECTION. It's not possible for this:
4^x = -1
So you simply ..REJECT!

The third question is more of an "INSPECTION" and there are other ways to prove it.
So this question is more of a check than a proof.
..4^x-9^x = 0

We got the same exponents ..

..(2^2)^x-(3^2)^x = 0
2x - 2x = 0
x = 0
If we apply that..
..4^0 - 9^0 = 0
1-1 = 0
0=0
//

SLIDE 8/10

Then we started to look over the graphing world of exponential graphs!
I personally found it interesting!
We were first asked to graph what you see on this slide.

SEE THE PATTERN?! .. AWESOME... AWESOME POSSUM =)

SLIDE 9/10

These are how the graphs look like!
y = 0 is the asymptote.
It will never touch the X axis. EVERY GRAPH HAS THE POINT (1,0)
We tried things such as a different coefficient and vertical and horizontal shifts.
We predicted how the graph would move according to the coefficient.
MASTER JABBAMATHEE predicted that the graph will slowly get bigger, after the point (1,0) it will quickly get bigger. As seen with the graph:
f(x) = 5^x
compared to f(x) = 2^x

SLIDE 10/10

Now we fiddle with our calculators!
We looked at Y = (-2)^x
REMEMBER, BRACKETS!

Then pressed ZOOM - and pressed "4"

Which gave us the graph you now see on the slide.

Sometimes there isn't a Y value there. Why? Because it doesn't exist. (IMAGINARY NUMBER)
.. The equation (-2)^x ..
As you see from the 2 examples. .6 and .5 Let's test them:

(-2) ^ (6/10)
= (-2)^(3/5)
Which allows the answer to be negative and existing. Meaning there is a Y value.

(-2)^(5/10)
= (-2)^(1/2)
.. IMPOSSIBLE! UNREAL ANSWER.

That's all we did for today!

HOMEWORK WAS EXERCISE 19 AND 20.
INCLUDING THE QUESTIONS THAT WE DID NOT DO BEFORE!

THIS IS NUMBER ELEVEN, signing out.