Exponential Growth and Decay in the real world

Hi guys miss joseph here, thank you for joining me

for a lesson on solving exponential.

Okay so the first few slides, more than a few,

I'm going to review um the properties of exponents simplifying

exponential expressions sketching grass of exponential functions.

So those of you who didn't take notes on the

previous videos, you should write the notes down now please.

And what's new is use the properties of exponents to

solve the exponential equations.

And those of you who have been writing down the

notes stay till the end but I'll try to go

fast here.

Okay so we need to talk about the difference between

a polynomial function and an exponential function.

A polynomial function has the basic form F of X

equals X to the power of three.

For example an exponential function on the other hand has

the basic form F of X equals three to the

power of X.

Can you see the difference?

An exponential function has the variable in the exponents not

in the base.

And the general form is into the power of X.

And N.

Is greater than zero.

Okay. Why study exponential functions?

Exponential functions are used in our real world to measure

growth interest in decay.

So here's some examples of growth exponential functions rumors.

If you start one rumor, you tell one person something

they promised not to tell.

They tell another person who tells two more people.

And boom, you're rumors have that rumor?

One rumor has grown exponentially.

Um The human population grows exponentially bacteria, computer technology, nuclear

chain reactions, compound interest decay.

Examples of decay are The number of friends are going

to have if you keep spreading those rumors from the

growth. Also carbon 14 dating half life.

And Newton's law of cooling.

Okay, here's the laws of exponents.

If you didn't write these down, pause me write them

down. Here's properties of exponents.

So when we multiply and we have the same base

we add the exponents but keep the base.

So two to the power of two times.

To the power of three is to the power of

five. Okay When you um when you add like two

plus negative six on the next example and you get

negative four a negative exponents does not make the number

negative, it makes the number the opposite.

So a reciprocal of a fraction.

So 22 negative four.

It would be 1/2 to the fourth power which would

be one to the 1/16.

And then power to a power to to the third

times. To our sorry hard to empower.

Two to the third to the power of two is

equal to two to the sixth which is 64.

Yeah. Okay this is a tricky 13 over to over

three to the power of negative three.

There's parentheses around that fraction.

That means that the negative three gets distributed to both

the numerator and the denominator.

So we end up with two to the power of

negative 3/3 to the power of negative three.

And then those flip where they're at.

So three to the power of three flips to the

top And it becomes a positive power and two to

the power of three flips the bottom and it becomes

a positive power.

So we have 27/8.

And then when you're dividing 3/7 over three to power

7/3 to the power of nine subtract.

So seven minus nine is negative to negative bumps into

the bottom 1/3 squared is nine.

Okay. Here's when we might not have talked about um

2 to 1 half power times eight to the one

half power.

You can multiply the two times eight and keep it

to the one half power.

And you get 16 to the one half power and

uh one half power really just means square root.

So the square is 16 is for okay.

An exponential function is just like any other function.

It has the input, that's the X values, the output,

that's the Y values.

And so here is the function to to the power

of X.

Remember anything to the power of zero is one When

you take away all powers, you just have yourself left.

Yeah, this is a good thing.

Okay. And we made tables for two to the power

of X.

If you did not copy that down on previous videos,

then you should do that now.

And we did graphs of two to the power of

X. Same thing.

Cause me copy it down if you didn't.

Yeah. Okay, we did a table for three to the

power of X.

Can you guys hear that stellar jay?

He's right above me.

And we did a graph of three to the power

of axe.

If you did not write those down, you should write

them down.

Okay, then I over laid to to the power of

X and three to the power of Acs because I

wanted to show you the bigger than number, the faster

it grows.

So too to the power of X is in blue,

three to the power of X is in red and

if you notice the red one goes up steeper.

So in other words, a larger base means the graph

rises more quickly and the smaller base means it rises

more gradually.

Okay, and then we've talked about decay functions, um, which

would be a fraction that's between zero and one.

Um So one half to the power of X.

If you didn't write that down you should know.

And here's the graph.

So it's a decay function.

It starts up super high and shoots down really low.

Uh huh.

Okay. Here we go.

This is what's new solving exponential equations with the same

base. So if we have B to the power of

M equals B to the power of N.

We have the exact same base.

We have an equals between them.

Take care of the basis.

Make the exponents equal.

All right so five to the power of X equals

1 25.

Okay? Um Five to the power of three is 1

25. So X would be three.

Okay, solve this one X seven to the power of

X minus one equals seven to the power of negative

one half.

So same base take away the bases.

Set the exponents equal to each other and salt.

So X minus one equals negative a half, add one

to both sides of X equals a half.

Okay here's another example to to the power of X

plus six equals two to the power of five.

Same base take away the basis exponents are equal to

each other.

You should be copying down these examples please.

So pause me.

I know I'm going through them quickly but pause me

and write these examples down.

Okay so three to the power of two, X minus

five equals three to the power of seven.

Take away the bases make the exponents equal to each

other so two X minus five equals seven.

Okay now you try this one copy this down, pause

me and come back and check your answer.

Did you do it?

Did you get x equals negative five.

If you did you go You if you didn't please

pause this and check and see what you did wrong.

Okay what if the bases are not the same?

Then we try to make them the same.

So eight to the power of X equals to eight

can be divided by two and two to the power

of three equals eight.

So what I did is I change the eight tattoo

to the power of three and I put that in

parentheses and the power of X is still there because

it was it was on the part of it was

eight to the power of X.

So I put two to the power of three.

All to the power of X equals two to the

one. Power.

Power to a power.

We multiply so two to the power of three, X

equals two to the first power.

And so then you set the power is equal to

each other.

So we get X equals a third.

Okay, eight to the power of X equals four.

Those are both divisible by two.

And we can do powers of two to make eight

and 42 to the power of three is eight.

I put that in parentheses to signify that's what represents

the 82 to the power of to is the four

distribute the three and the X.

So you get two to the three X equals to

two. Set them equal to each other and divide copy

these down please.

Next two examples nine to the power of two, X

minus five equals 27.

Those are both divisible by three and We can use

powers of three to get to nine and 27.

So three squared is nine.

I put that in parentheses Because that represents the nine

and three Q just 27.

Okay look Because my original power was 9 to the

power of two X -5.

Now I still have a power to a power three

squared to the power of to explain.

Is five is still power to a power and you

still multiply and that two gets distributed.

So three to the power of two to the power

of two x minus five becomes three to the power

of two times two is four X minus two times

five is 10 equals three to the power of three.

Then set your powers equal to each other and solve

and you don't get a pretty number.

You get 13/4.

Okay. Next 1 25 to the power of two X

minus one equals 1.

25 to the power of three x plus four.

Both of those are divisible by five.

And yes we have powers of five that make both

of those numbers.

So five squared represents the 25 5 cube represents the

1 25.

We have power to a power now I'm going to

multiply the two powers.

So two times two X plus one becomes four X

plus or minus sorry minus two and three times three

X plus four becomes nine X minus 12.

And then you set the powers equal to each other

four X minus two equals nine X minus 12 solve.

And you get pretty number X equals two.

You sure you're writing this down?

Okay write this down and you try this one then

come back to me.

Did you do it?

Did you get a crazy number 19/6.

Okay so we changed 4 to 2 to the power

of to 8 to 2 to the power of three.

Then you distribute two times three X plus five becomes

+66 plus 10 and three times four x minus three

becomes 12 X minus nine.

Set six X plus 10 equals 12 X minus nine.

And solve.

And you end up with 19/6.

Copy that down please.

Okay, try this one.

This one is so tricky, pause me see if you

can figure it out and then come back to me.

Okay, here's the tricky part.

1/27 can also be written as 27 to the power

of negative one and three to the power of three

is 27.

So multiply that power of three times the power negative

one and you get three to the power of negative

three replaces the 1/27 And three squared replaces the nine.

Power to power.

We multiply.

So two attempts to X plus three becomes four X

plus six negative three times three X plus one becomes

negative nine X minus three solve.

And we get another Not very pretty number negative 9/13.

Okay now you try this one please pause me and

come back and see if you got it.

Did you do it did you get nine over to?

Okay so three to the power of two X minus

five equals 81.

Okay. Three to the power of four equals 81.

If you don't know your powers just mess around on

the calculator.

Um and see if you can come up with it.

It's it's for these cases it's gonna be a one

of the smaller numbers um like 2345.

Um Okay so then we set the powers equal to

each other and we solve and we get nine over

to. Okay now one last one this one is very

tricky. Look I have 1/7 equals 49 to the power

of x minus five times seven to the power of

x minus nine.

This is so tricky.

See if you can pause me and see if you

can figure this out on your own.

Okay so 1/7 can be written as seven to the

negative one 49 can be written as seven squared.

And so then we have seven squared, raise the power

of x minus five and seven raise the power of

x minus nine.

Well if you remember when you have the same basis

and you're multiplying all you do is add the exponents.

So what you're doing is distributing the two.

So you get two x minus 10 and then you're

adding it to x minus nine.

And when you combine like terms you end up with

X equals six.

Whoa! Mind alone, pause me copy that down, take that

in and that was a really awesome problem.

Okay And the next level math you're gonna learn how

to solve exponential functions using what's called logarithms.

Um If the bases aren't the same and they can't

be made into the same basis, but we're stopping there,

we're gonna stop with what you can make into the

same basis.

Okay? Have a great day.

Thank you for sticking with me.

Exponential Growth and Decay in the real world

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