Let's get our feet wet solving these exponential equations. Now I'm going to
show you a way that's going to work for the, obviously for the problems I'm
showing you here. But I must be honest; it's not going to be the way we're going
to use at the end of the semester with logarithms and such.
Okay, don't let that scare you but I tell you that because I want you to pay close
attention to the reason we're doing this and that is to review the rules of
exponents.
Ok, the facts about exponential functions are: Number one, if we have something of
the form y equals B to the X, B has got to be more than zero, can't be negative.
If it was negative, it, it would yield a non-real answer because you'd have to
take a root of a negative number.
Okay that's not very useful, non-real answer so don't worry about b being
negative. That's for sure.
If B is zero, if B is zero, if you raise 0 to any power then y will always be 0, won't it?
So, if it never changes ,that's not very exciting. So we're not going to have that
happen either. b can't equal one.
Well, it could but if it did 1 to any power is always one, isn't it?
And that would be boring, so that's not going to change either. So you don't have to worry
about being one either. So basically we're going to look at the the base, b
being numbers larger than zero but not one. The property we're going to use
today and the most important fact about exponential functions is
if b to the x equals b to the y, (now, the bases have to be the same for this to work) but
if b to the x equals b to the y, then we know x equals y and we can use that to
solve equations and we're about to do it. Let's solve this now. We've got six to
the x equals 36. Obviously the bases are different. So let's create similar bases.
We're going to rewrite this equation somehow with common bases. Now you see how
the right side could be written with a base of 6. 36 is 6 to the twoth, isn't it?
So if we write it 6 to the x equals 6 to the twoth with common bases.
Well, now we can use our property.
Okay, x must be two if 6 to the x is 6 to the twoth, X must be two.
Let's try another. Now the hard part is coming up with that common base.
Let's see, look for common base. What would be a common base that we could
rewrite 64 and 1600? Practice makes perfect on these. I smell four this time
because 64 is four to the third, isn't it? And sixteen is four to the twoth but we
have to remind ourselves of the rules of exponents. If we rewrite 64 as four to the
third, the rules of exponents tell us that four to the third to the x
is, (well, we multiply exponents, don't we?) is four to the 3x. Now we've got each side
with a common base four to the 3x equals 4 to the twoth and 3x equals two. Now we're
not done yet. We, but we have lowered the x down to a base level, if you would, and
we can solve it by solving for x by dividing by 3 to both sides and x will
equal to two-thirds.
Ok, note how we're reviewing the rules of exponents and in fact let's do it
because we're going to review it as we check this answer. What is 64 to the
two-thirds power? Course we hope it's 16. Remember how to do this? Well, 64 to the
two-thirds is 64 to the one-third squared or 64 to the one-third times 64 to
the one-third. Recall that you add exponents. Now the one-third power,
what does that mean? Remember, one-third power means the cube root, the third root
So 64 to the one-third is really the cube root of sixty-four which is four.
And now we raise four to the twoth and we've checked it.
Ok, let's do another one. Recall that our method is to look for a common base. Now
what do you smell this time? Nine to the 2x + 1 and 81. You know you smell nine!
Let's rewrite 81 as 9 to the twoth. Now, if nine to the 2x plus 1 is 9 to the twoth
then 2x plus 1 must equal two. We still have to solve for x but we're well on
our way.
Let's subtract 1 from both sides
and then divide by two, both sides and we get x is a half, a half and that wasn't that
difficult.
Let's take that half. Let's check it, not because I said so but hopefully we can
use the rules of exponents to do this.
This one's not that difficult. If I replace x with a half, two times a half
is one and one plus one is two and nine to the twoth...It checks... is 81 and we've
got more exponent rules to review. So let's do another. This one looks a little
bit more difficult. It's got a reciprocal. Give you a hint 1 over 27 is 1 over 3 to
the third so the common base we're going to use is 3. 1 over 3 to the third. Isn't that
three to the minus 3rd? That's what negative exponents do. So now we have a
common base. 3 to the minus three equals three to the 2x and we can use our
property minus 3 must equal to x and solve for x by dividing by 2
and we'll get x equals minus three-halves. Now, dare we take minus
three-halves and put it back in and see if it works? See
if it equals one 27th. So instead of X, I'll put minus three-halves. Now two x minus
three halves is minus 3 and as we said before three to the minus 3
power that means reciprocate 3 to the positive 3 power. So three to the minus 3
is 1 over 3 to the 3rd which is one 27th. How about that?
Let's do another one. Practice makes perfect.
Ok, we're going to look for common base. Now, look at this. What common base would
you use here? What number can you raise to a power and get 32 and also get four?
Well it's two! 32 is 2 to the fifth and of course 4 is 2 to the twoth. We got to
recall our rules of exponents here because we still don't have exactly a
common base. What are we going to do with that five and that x? The rules of
exponents tell me that I have to multiply. In this case I have 2 to the 5x
equals 2 to the twoth and now I can use my, my property and then solve for x by
dividing by 5 and X is two-fifths. Of course we're gonna have to check it but
don't, don't worry we can do this. Let's take X and replace it with our answer,
two fifths, and recall how to do this too. A little bit at a time. Okay, we'll do
32 to the one fifth and you know square it. 32 to the one-fifth squared is 32 the
two fifths, ok.
32 to the one-fifth. What does that mean?
32 to the one-fifth means the fifth root of 32. Now you need to understand what is the
fifth root of 32? What number, to the fifth power equals 32? I may be spoiling
you a little bit but it comes out even. It's two.
Okay, now I'll take that two, raise it to the twoth power and it looks like it
works. I get the other side. I get the four. Okay I don't suppose you're
noticing how I'm sneaking in these little factoids about exponents such as
negative exponents and fractional exponents.
Well, I have another for you. Let's look at this multiple term exponent 125 and 25.
Now take a good sniff and tell me what's going to be the common base here?
Hopefully practice has made perfect and you smell, you smell five. Let's rewrite 125
as five to the third and 25 as 5 to the twoth. Now we'll have to use our rules of
exponents to kind of simplify the left side.
I'm going to multiply the 3 times the X minus 2 and get five to the 3x minus six.
Now I can use my principal because I actually have common bases and just
rewrite that 3x minus 6 equals two.
Now let's solve for x by adding 6 to both sides and dividing by 3 and we'll
get
x is, x is eight-thirds. Now,
another way to check this
course plugging in eight thirds is to use, to correctly use your calculator. Now
it's important to note that if you type a hundred and twenty-five to the eight
thirds minus 2you better always put parentheses around the complete, the
complete exponent. Always put parentheses around the exponent, ok, to force the
order of operations that you want and, son of a gun, if you do it correctly you
will get 25 and it does check. But that's the thing I wanted to remind you of; it's
a very common mistake. People forget to put parentheses around exponents. If we
didn't put parentheses there, hope you appreciate that would raise 125 to the
eight thirds power, get an answer and then subtract two from that and that's
not what we want here.
Ok, let's go try and practice these. Look for that common base. Go do the homework.