Now we're going to take everything we've learned about logarithms up to this
point, put it together and use it to solve some very complicated equations
involving logarithms.
Do you remember,
remember the idea of contracting multiple logs into one log using the
properties of logs? Now we use a couple of them here. For instance, we're going to
take this expression and make it into one logarithm using a couple of the
properties. Remember where am I allowed to put that two other than in front? Using
the same property, I'm going to place three
and what becomes, when we're contracting, of addition?
Addition becomes
multiplication.
We don't have it here but we can... remember, hopefully, that subtraction
becomes division and anyway our goal here was to make this into one big fat
logarithm. 3 to the second is 9. 2 to the third is eight and I believe that we've
done it. Well, hold on just a second. I'll show you why you might want to make this
into one big logarithm. Another thing that we learned regarding equations was
to switch it into exponential form. Remember what the base is. Remember that
the log is an exponent. So we could rewrite this using the old switcheroo as
the base four to the exponent that's what the logarithm is to the exponent 2
equals the number or expression X minus 2. Well, four to the second is 16 and now we
can solve the equation adding 2 to both sides.
X in this case is 18. Now, it's very important; hopefully you learned from
the last homework that you have to put it in and check it because they don't
always work. We take our 18 and put it in. We'll take the log base 4 of 18-2 or 16.
Now the log base 4 of 16 is that two. Well, four to the twoth does equals 16 so
it checks. But make sure you always check as often some of these don't work.
Using those two principles, the properties of logs to condense and the
old switcheroo,
let's look at a more advanced logarithm equation.
Now we can't use the old switcheroo here, can we,
until we only have one logarithm. So we're going to have to condense it into
one logarithm; then we can use the old switcheroo.
So let's do that. I condensed addition into multiplication and now I have one
logarithm. Hopefully you see the need for that and I can now change it using the
old switcheroo. Two to the one, the base of 2, the exponent or logarithm of 1 equals
the number or expression x times quantity X minus 1.
Well, now I can solve it to the 1 this 2 and x squared minus x is going to turn
this into a quadratic equation which I can solve if I get a zero on one side.
I'll do that.
This one is factorable so the answers, uh, there's two, are going to be...
What makes that zero a positive 2 and what makes that 0 is a negative one.
So we have two...I should say possible answers because we need to plug these in because
we're not sure they work. Well, let's start with, let's try putting in negative 1.
If you put in negative 1 for x you come up against the wall right away because
you have to ask yourself...remember the rules of logs... you can't take the log of
a negative number so that has got to be out. Now let's try two; he may be out as
well. We put a 2 in for x, the log base two of two log base two of two is what?
Is one and the log base 2 of 1.
What's the log of one no matter what base you use and, son-of-a-gun, the left
side will be 1 plus 0 or 1 and the right side is also one and this one works. So
the one solution to this is two. So remember to always plug in and check
your answers. Got the idea?
Well, go try it. Do your condensing; then do your old switcheroo and solve those
equations.