Example 2 Solve
each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
Okay, so we say above that if we had a logarithm in front
the left side we could get the x
out of the exponent. That’s easy
enough to do. We’ll just put a
logarithm in front of the left side.
However, if we put a logarithm there we also must put a logarithm in
front of the right side. This is
commonly referred to as taking the
logarithm of both sides.
We can use any logarithm that we’d like to so let’s try
the natural logarithm.
Now, we need to solve for x. This is easier than it
looks. If we had then we could all solve for x simply by dividing both sides by
7. It works in exactly the same manner
here. Both ln7 and ln9 are just
numbers. Admittedly, it would take a
calculator to determine just what those numbers are, but they are numbers and
so we can do the same thing here.
Now, that is technically the exact answer. However, in this case it’s usually best to
get a decimal answer so let’s go one step further.
Note that the answers to these are decimal answers more often
than not.
Also, be careful here to not make the following mistake.
The two are clearly different numbers.
Finally, let’s also use the common logarithm to make sure
that we get the same answer.
So, sure enough the same answer. We can use either logarithm, although there
are times when it is more convenient to use one over the other.
[Return to Problems]
(b)
In this case we can’t just put a logarithm in front of
both sides. There are two reasons for
this. First on the right side we’ve
got a zero and we know from the previous section that we can’t take the logarithm
of zero. Next, in order to move the
exponent down it has to be on the whole term inside the logarithm and that
just won’t be the case with this equation in its present form.
So, the first step is to move on of the terms to the other
side of the equal sign, then we will take the logarithm of both sides using
the natural logarithm.
Okay, this looks messy, but again, it’s really not that
bad. Let’s look at the following
equation first.
We can all solve this equation and so that means that we
can solve the one that we’ve got.
Again the ln2 and ln3 are just numbers and so the process is exactly
the same. The answer will be messier
than this equation, but the process is identical. Here is the work for this one.
So, we get all the terms with y in them on one side and all the other terms on the other
side. Once this is done we then factor
out a y and divide by the
coefficient. Again, we would prefer a
decimal answer so let’s get that.
[Return to Problems]
(c)
Now, this one is a little easier than the previous
one. Again, we’ll take the natural
logarithm of both sides.
Notice that we didn’t take the exponent out of this
one. That is because we want to use
the following property with this one.
We saw this in the previous section (in more general form)
and by using this here we will make our life significantly easier. Using this property gives,
Notice the parenthesis around the 2 in the logarithm this
time. They are there to make sure that
we don’t make the following mistake.
Be very careful with this mistake. It is easy to make when you aren’t paying
attention to what you’re doing or are in a hurry.
[Return to Problems]
(d)
The equation in this part is similar to the previous part
except this time we’ve got a base of 10 and so recalling the fact that,
it makes more sense to use common logarithms this time
around.
Here is the work for this equation.
This could have been done with natural logarithms but the
work would have been messier.
[Return to Problems]
(e)
With this final equation we’ve got a couple of
issues. First we’ll need to move the
number over to the other side. In
order to take the logarithm of both sides we need to have the exponential on
one side by itself. Doing this gives,
Next, we’ve got to get a coefficient of 1 on the
exponential. We can only use the facts
to simplify this if there isn’t a coefficient on the exponential. So, divide both sides by 5 to get,
At this point we will take the logarithm of both sides
using the natural logarithm since there is an e in the equation.
[Return to Problems]
