In this section we’ll take a look at solving equations with
exponential functions or logarithms in them.
We’ll start with equations that involve exponential
functions. The main property that we’ll
need for these equations is,
Example 1 Solve
The first step is to get the exponential all by itself on
one side of the equation with a coefficient of one.
Now, we need to get the z out of the exponent so we can solve for it. To do this we will use the property
above. Since we have an e in the equation we’ll use the
natural logarithm. First we take the
logarithm of both sides and then use the property to simplify the equation.
All we need to do now is solve this equation for z.
Example 2 Solve
Now, in this case it looks like the best logarithm to use
is the common logarithm since left hand side has a base of 10. There’s no initial simplification to do, so
just take the log of both sides and simplify.
At this point, we’ve just got a quadratic that can be
So, it looks like the solutions in this case are and .
Now that we’ve seen a couple of equations where the variable
only appears in the exponent we need to see an example with variables both in
the exponent and out of it.
Example 3 Solve
The first step is to factor an x out of both terms.
DO NOT DIVIDE AN x FROM BOTH TERMS!!!!
Note that it is very tempting to “simplify” the equation
by dividing an x out of both
terms. However, if you do that you’ll
miss a solution as we’ll see.
So, it’s now a little easier to deal with. From this we can see that we get one of two
The first possibility has nothing more to do, except
notice that if we had divided both sides by an x we would have missed this one so be careful. In the second possibility we’ve got a
little more to do. This is an equation
similar to the first two that we did in this section.
Don’t forget that !
So, the two solutions are and .
The next equation is a more complicated (looking at least…)
example similar to the previous one.
Example 4 Solve
As with the previous problem do NOT divide an out of both sides. Doing this will lose solutions even though
it “simplifies” the equation. Note
however, that if you can divide a term out then you can also factor it out if
the equation is written properly.
So, the first step here is to move everything to one side
of the equation and then to factor out the .
At this point all we need to do is set each factor equal
to zero and solve each.
The three solutions are then and .
As a final example let’s take a look at an equation that contains
two different logarithms.
Example 5 Solve
The first step here is to get one exponential on each side
and then we’ll divide both sides by one of them (which doesn’t matter for the
most part) so we’ll have a quotient of two exponentials. The quotient can then be simplified and
we’ll finally get both coefficients on the other side. Doing all of this gives,
Note that while we said that it doesn’t really matter
which exponential we divide out by doing it the way we did here we’ll avoid a
negative coefficient on the x. Not a major issue, but those minus signs on
coefficients are really easy to lose on occasion.
This is now in a form that we can deal with so here’s the
rest of the solution.
This equation has a single solution of .
Now let’s take a look at some equations that involve
logarithms. The main property that we’ll
be using to solve these kinds of equations is,
Example 6 Solve
This first step in this problem is to get the logarithm by
itself on one side of the equation with a coefficient of 1.
Now, we need to get the x out of the logarithm and the best way to do that is to
“exponentiate” both sides using e. In other word,
So using the property above with e, since there is a natural logarithm in the equation, we get,
Now all that we need to do is solve this for x.
At this point we might be tempted to say that we’re done
and move on. However, we do need to be
careful. Recall from the previous section that we can’t plug a negative number
into a logarithm. This, by itself, doesn’t
mean that our answer won’t work since its negative. What we need to do is plug it into the
logarithm and make sure that will not be negative. I’ll leave it to you to verify that this is
in fact positive upon plugging our solution into the logarithm and so is in fact a solution to the equation.
Let’s now take a look at a more complicated equation. Often there will be more than one logarithm
in the equation. When this happens we
will need to use on or more of the following properties to combine all the
logarithms into a single logarithm. Once
this has been done we can proceed as we did in the previous example.
Example 7 Solve
First get the two logarithms combined into a single
Now, exponentiate both sides and solve for x.
Finally, we just need to make sure that the solution, ,
doesn’t produce negative numbers in both of the original logarithms. It doesn’t, so this is in fact our
solution to this problem.
Let’s take a look at one more example.
When solving equations with logarithms it is important to
check your potential solutions to make sure that they don’t generate logarithms
of negative numbers or zero. It is also
important to make sure that you do the checks in the original equation. If you
check them in the second logarithm above (after we’ve combined the two logs)
both solutions will appear to work! This
is because in combining the two logarithms we’ve actually changed the problem. In fact, it is this change that introduces
the extra solution that we couldn’t use!
Also be careful in solving equations containing logarithms
to not get locked into the idea that you will get two potential solutions and
only one of these will work. It is
possible to have problems where both are solutions and where neither are