00:00 Good day, My name is Afeez. I'm going to take you further math and our today's topic is going

00:17 to be a continuation of indices, which is called Exponential Equations. At the end of this topic,

00:25 we should be able to know how to find Exponential Equation of Linear Form, Exponential Equation

00:31 of a Quadratic Form. And let's define what the Exponential Equation is. Equation in which

00:39 the unknown variable occurs as part of the index is called Exponential Equation. That is,

00:45 when you have an equation of the form, when you have the equation of this form, e raised

00:49 to power x equals b, then this is what we call Exponential Equation or we call it in this

00:56 share or in this share equation. So you can see that the index is unknown. The power is unknown.

01:10 And the base, we have a base and we have equals to. So when you have this is an indices and

01:19 why this is what? Making it to be an equation. So when you have an expression in which there's

01:24 equal to sign in between them, then we have an equation. So we have a x, a raised to power

01:30 x equals b is an exponential equation. So example of a linear in this, our linear in this

01:40 share equation. So we have two raised to power x equals to two, one over two raised to power

01:48 x equals eight. Then we also have 0.25 raised to power x plus one equals 16. So all these are

01:55 what? They are linear. They are linear. If we are able to find the value of our x in which,

02:01 by the time the two, the base are equal, then we can have a linear form. So Exponential Equation

02:10 of Linear Form. Under the Exponential Equation of Linear Form, you must note that the base,

02:17 if the base are the same, then you must have the, we have, if for example, you have a

02:25 raised to power x equals a raised to power one, you can see that we have the base, the base

02:31 are the same thing. So our x now can be once, can be one. So this is what we are saying

02:35 that when you have an exponential equation of linear form, you must take note that our

02:42 base must be the same before you can find the value of your x. So let's take a look to

02:49 this example. We have, we solve two raised to power x equals 32. We have two raised to

02:59 power x equals 32. So and we want to find the value of x. Meaning that these 32 must be

03:10 in the form of two raised to power x. So we have two raised to power x equals two raised

03:15 to power five. Two raised to power five will give us 32. So we have, and you can see that

03:21 we have the same base. Since we have the same base, x will not be worth five. So therefore,

03:27 the value of x is what is five. Also, let's take a look to this, to this example as well.

03:37 We have zero points zero point two five all raised to power x plus one equals 16. And don't

03:50 forget, we say that our base must be the same thing before we can find the value of x. So

03:56 it means that if we have, how do we make this one to have the same base with the right-hand

04:01 side? So we have, this is going to be 25 all over 100. That is 0.25 give us 25 over 100

04:10 all is power of x plus one equals. And this is, we give us four is about two. Now you see

04:18 that there 25 goes there one 25 goes there for. So it means we have one over four all raised

04:25 to power x plus one equals four raised to power two. And as you can see, you see that there

04:35 we can write this. We can have four one over four in in this form one four raised to power

04:44 minus one. That is four is per minus one. All raised to power was x plus one equals four

04:54 raised to power two. And by the time we use what we call the power law in our indices, in our

04:59 law of indices, we can easily have this. We can easily have this. So this is going to be four,

05:10 it's going to be four raised to power, it's going to be four raised to power minus into x plus one

05:16 equals four raised to power two. You can see that we have the same base. Then since we have

05:23 the same base, it's going to be minus x plus one equals two. And we can hope on the brackets

05:30 with this we have minus x minus one equals two. And this would be minus x, we collect the

05:39 like times, we have minus x equals two plus one. Our minus one will be called plus where we

05:46 transfer this to the other side. So we have minus x equals three. And therefore we have

06:01 x equals minus three. So therefore, our x is equals minus three. We have two raised to power two

06:10 x, our minus six raised to power two raised to power x plus eight equals zero. So we can have

06:18 two raised to power x or raised to power two. We use what we call the power law yeah. So minus six

06:25 raised to power two raised to power x plus eight equals zero. So let's why equals two raised to

06:34 power x. So it means wherever we see two raised to power x in this equation in the above equation

06:40 we put y. So it means we have y squared minus six y plus eight equals zero. Then we cannot use

06:50 what we call factorization method to solve this quadratic equation. So that is y squared minus four y

06:58 minus two y plus eight equals zero. So we can say y is common yet we have y into y minus four.

07:10 Then two is common minus two is common here we have minus two into bracket y minus four equals zero.

07:19 We can see that one minus four is common between these we have y minus four

07:26 into y minus two or equals zero. So it means y minus four equals zero or y minus two equals zero.

07:36 So therefore we have y equals four or two. Now we can

07:47 when y equals four. Don't forget that we said later y equals two raised to power x. So we

07:57 can easily say that two raised to power x equals four. Then this is two raised to power x equals two

08:05 raised to power two. Then since we have the same base therefore x equals two. That is when our y equals four.

08:16 So when y equals two. It means we have two raised to power x equals two. And that is

08:24 two raised to power x equals two raised to power one. And since we have the same base we have

08:29 y for x equals one. So it means x equals one or two.

08:39 So we have this as a size. This as number one is for the exponential equation

08:56 of linear of quadratic form. And that of x or x is number two is for

09:01 exponential equation of a linear form. So therefore we can use this the method that we use

09:11 in our previous examples to solve this as a size. So thanks for watching. Once again my name is

09:18 Bistam Fis. See you next class.