Master Solving Exponential equations without using a calculator

Master Solving Exponential equations without using a calculator

Welcome ladies and gentlemen. So what I’d like
to do is show you how to solve an
exponential equation using the one to one property. Now what’s special about
the one to one property is majority of the times, we
don’t have to use a calculator. All of these problems
I’m going to solve, all these exponential
equations I’m going to solve, I’m not going to
use a calculator. Now we do have to have
certain equations, because the next video
that I’m going to make is solving them with
using a calculator. So you can use the
one to one property for every single problem. But for all of these we can. And again to kind of remind
you of the one to one property, it’s basically
just saying if you had a to the x equals a
to the y, then x equals y. Now that’s your kind
of formal definition. And the way I just kind of
think about it– there’s my calculator– is to think
about it in real numbers. If I had 3 squared is
equal to 3 to the x, then what does x have to equal? If both sides are
equal to each other and we know one power’s
2, the other power’s x, what does x have to be? Well, it’s obvious 2 has to
equal x or x has to equal 2. So that’s basically the
one to one property. And a lot of times, sometimes
we’ll just say they cancel out and delete them. But what it means
is when you have the bases are exactly the
same, then their powers are equal to each other. So what happens with this is I
didn’t do any basic ones, which I guess I probably could have. But you look at 4
to the x equals 16. Well in this case,
we don’t have bases that are exactly the same. So therefore what I’m going
to need to do in this case is I’m going to need to rewrite
this expression with the bases being exactly the same. And the reason why the
one to one property works for all of these is
because we can rewrite these all with having the same base. So you gotta look at 16
and say, can I rewrite 16 as 4 raised to a power? Well this one should
be fairly obvious. We have 4 to the x. And I could write 4
squared as equal to 16. So if I have 4 to the
x equals 4 squared, well now I can just say
that x is equal to 2. And there I go, done. Now in this next
example, we know that 4 square is equal to 16. But we have a problem. This 16 is in the denominator. So here I have an exponent. And here I have a fraction. So one of the things
that we need to do is learn to rewrite fractions
with negative powers. And that comes into one of our
rules from negative exponents. I’ll write the one to one
property here as well. So I can rewrite 16
with a negative power. 4 to the x equals 16 to
the negative first power. And again, that comes from
our rules of using negative x. Now I know that 16 I can
rewrite as 4 squared. Just make sure that you’re using
your parentheses correctly. Now you’re going to want to use
the power rule, which remember, the power rule is when you
have an exponent raised to another power. You multiply the powers. So now I’m going to multiply
by 2 times negative 1. So 4 to the x equals
4 times a negative 2. Now I can say that x
is equal to negative 2. So there’s a difference
when x equals 2 or x equals negative 2. So now we have an
exponent that’s under– now we have a fraction
that’s being raised to the x. Well again the same
thing, we’re going to want to use our
negative exponent role. So I’m going to
rewrite this as 2 to the negative first
power times x equals 32. I multiply these,
2 the negative x. Now again I want
to think about 32. Now a lot of times people
sometimes kind of get confused. What we want to do is we know
we can’t rewrite 2 as a base 32. So we want to rewrite
32 as a base 2. So you always want to kind of
go back to your smaller base. And so you’re trying to see,
can I take that smaller base and raise it to a power
to get to my other base? So a lot of times what I
just tell students to do is just to the side, just start
writing down what each of them equal. 2 to the first power is 2. 2 squared is 4. 2 cubed is 8. 2 to the fourth power is 16. 2 to the fifth power is 32. So therefore I know
32 is equivalent to 2 to the fifth power. Now I have my one
to one property. So I can say negative
x is equal to 5. Just divide by a
negative 1 on both sides. x equals a negative 5. In the next example,
you can see now we have a fraction on both sides. And again, but we want to kind
of look at– fractions a lot of times, students
just freeze up. They see a fraction,
they freeze up. But again, we can
break down the fraction into the numerator
and the denominator and see how the numerator
and the denominator are related on both sides. For my numerators, you can see
just 2 to 4 is being squared. From 3 to 9 is just
being squared, right? So all I can do is I
can rewrite this as 2/3 to the x equals 2/3 squared. Because remember when
you’re squaring a fraction, you square the numerator and
you square the denominator. Or whenever you have a
fraction raised to a power, that power distributes to
the numerator as well as the denominator. So therefore now they
have the same base, and I can say x is equal to 2. And that rule always
looks like this, a over b to the m equals a to
the m over b to the m. So those are kind of
like the simple ones, just kind of looking at. Usually what you’re going
to do is find the lower base and then rewrite
to the other base. But now what we’re
going to do is going to get into
some problems that are going to be a little
bit more difficult where the base is not
so much apparent. Because in this case, I
saw my lower number was 4 and I said that’s my base. Can I rewrite 16 as a base 4? And we could. Well here my lower base is 9. Can I rewrite 27 raised
to the ninth power? And I think crap, no I can’t. 9 squared is 81. So I’m kind of stuck there. So now what I need to do
is think about a lower base that I can raise to give me 9,
as well as a lower base that can give me 27. Well automatically when I
see 9 I think of 3 squared. So then I say all
right, well if I know 3 squared, 3 to the first is 3. 3 squared is 9. Hopefully you know
what 3 cubed is, but if you don’t, you’ll figure
out that it’s going to be 27. So therefore I actually
can rewrite both of these with the same base 3. Whereas this one is
going to be 3 squared x. And this one is 3 cubed. Actually I don’t really
need my parentheses. Now I apply my power rule. So I get 3 to the
2x equals 3 cubed. Therefore I have
2x is equal to 3. Divide by 2, divide by 2. x equals 3/2. In this next case,
now we just have an exponent that’s not just–
before I’ve been using just x. And it’s been pretty
simple, right? But now we can also have
expressions that are exponents. But it’s not really
going to matter as much. Again, in this case
I see 2 and I see 8. I know that I can raise
2 to the third power and that’s going to give me 8. So I just have 2 raised to
the 2x minus 1 equals 2 cubed. Now again, whenever your
bases are exactly the same, the powers are just
equal to each other. So now I have a
two step equation that I can just solve based
on my understanding of solving equations. I’m getting a lot of answers
that are 2, aren’t I? And there you go. Now in this case again
we look at 125 and 5. And automatically when I see
25, I think oh OK, well again, looking for the change of
base, not every problem that you do solving
equations is going to be using the change of base. But any time you
see a square number, think to use the one
to one property first. I see 25. I think 5 squared. That’s a square number. Now is 125, can I also take
5 and raise it to a power and give me 125? Yes, it’s going to be 3. So therefore I’ll rewrite this
as 5 cubed times 2x plus 1 equals 5 squared. Now to multiply these,
I have to make sure that I apply the
distributive property. Because remember
using the power rule, you’re applying the
distributive property. Bases are the same. So when I apply the
distributive property, I’m going to distribute. So therefore I’ll have 6x plus
3 equals 2 minus 3 minus 3. 6x equals negative 1. Divide by 6, divide by 6. x equals a negative 1/6. That doesn’t look good. And then last but not
least, we could also have our powers be
expressions on both sides. Again I look at 8 and 16 and I
think of 16 as being 4 squared, right? But the problem is I can’t
write 4 raised to a power to give me 8. Because if you
think about it, you have 4 to the first
power, which is 4. 4 squared is equal to 16. So we’re kind of stuck. We know 16 is 4 squared. But we can’t write
8 as 4 to a power. So we have to look
at a smaller number. And then fortunately I used
a lot of these to be simple, would be 2. Notice how 2 raised to a power
gives us 16 as well as 8. So that’s what I’m
going to want to use. And again ladies and gentlemen,
the main important thing is I would recommend
knowing 2, 3 at least to the first couple of
powers, because when you’re using the one
to one property, most of the problems in the book
or you’re seeing on tests, you’re using very
similar numbers. Even when it’s a fraction, the
numbers in the top and bottom are going to be square
numbers raised to a power. So therefore 2 to the
eighth is 2 cubed, times x plus 3 equals
2 to the fourth power, times– oops– 2 to the
fourth power times x minus 1. Now I need to apply the
distributive property. So it’s 2 to 3 times
x plus 1 equals 2 to the 4 times x minus 1. Now I can get rid of my bases. And I’m left with 3x
plus 3 equals 4x minus 4. Now I just need to
simply solve for x. So I’m going to get the
x’s on the same side. I’m going to get 7
equals x or x equals 7. So there you go
ladies and gentlemen. That is how you solve an
exponential question using the one to one property. Thanks.

6 thoughts to “Master Solving Exponential equations without using a calculator”

  1. Actually your answer to the last problem is wrong. The answer should be 13 instead of 7. You made a mistake by writing 3(x + 1) instead of 3(x + 3).

  2. I’m in grade 9 doing a Cambridge system and i always get confused with these equations and this video was extremely helpful. Thank you so much

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