All AboUt lEsSon

Solving Exponential Equations

Let’s start off by looking at the simpler method. This method will use the following fact about exponential functions.

Example 1 Solve each of the following.

(a)

(b)

(c)

(d)

Solution

(a)

In this first part we have the same base on both exponentials so there really isn’t much to do other than to set the two exponents equal to each other and solve for x.

(b)

Again, there really isn’t much to do here other than set the exponents equal since the base is the same in both exponentials.

In this case we get two solutions to the equation. That is perfectly acceptable so don’t worry about it when it happens.

(c)

Now, in this case we don’t have the same base so we can’t just set exponents equal. However, with a little manipulation of the right side we can get the same base on both exponents. To do this all we need to notice is that

. Here’s what we get when we use this fact.

Now, we still can’t just set exponents equal since the right side now has two exponents. If we recall our exponent properties we can fix this however.

We now have the same base and a single exponent on each base so we can use the property and set the exponents equal. Doing this gives,

So, after all that work we get a solution of

(d)

In this part we’ve got some issues with both sides. First the right side is a fraction and the left side isn’t. That is not the problem that it might appear to be however, so for a second let’s ignore that. The real issue here is that we can’t write 8 as a power of 4 and we can’t write 4 as a power of 8 as we did in the previous part.

The first thing to do in this problem is to get the same base on both sides and to so that we’ll have to note that we can write both 4 and 8 as a power of 2. So let’s do that.

It’s now time to take care of the fraction on the right side. To do this we simply need to remember the following exponent property.

Using this gives,

So, we now have the same base and each base has a single exponent on it so we can set the exponents equal.

Recall the following logarithm property from the last section.

Example 2 Solve each of the following equations.

(a)

(b)

Solution

(a)

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.

(b)

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.

Solving Logarithm Equations

Now, let’s start off by looking at equations in which each term is a logarithm and all the bases on the logarithms are the same. In this case we will use the fact that,

Example 1 Solve each of the following equations.

(a)

(b)

Solution

(a)

With this equation there are only two logarithms in the equation so it’s easy to get on one either side of the equal sign. We will also need to deal with the coefficient in front of the first term.

Now that we’ve got two logarithms with the same base and coefficients of 1 on either side of the equal sign we can drop the logs and solve.

Now, we do need to worry if this solution will produce any negative numbers or zeroes in the logarithms so the next step is to plug this into the original equation and see if it does.

Note that we don’t need to go all the way out with the check here. We just need to make sure that once we plug in the xwe don’t have any negative numbers or zeroes in the logarithms. Since we don’t in this case we have the solution, it is

(b)

Okay, in this equation we’ve got three logarithms and we can only have two. So, we saw how to do this kind of work in a set of examples in the previous section so we just need to do the same thing here. It doesn’t really matter how we do this, but since one side already has one logarithm on it we might as well combine the logs on the other side.

Now we’ve got one logarithm on either side of the equal sign, they are the same base and have coefficients of one so we can drop the logarithms and solve.

Now, before we declare these to be solutions we MUST check them in the original equation.

No logarithms of negative numbers and no logarithms of zero so this is a solution.

We don’t need to go any farther, there is a logarithm of a negative number in the first term (the others are also negative) and that’s all we need in order to exclude this as a solution.

Be careful here. We are not excluding

because it is negative, that’s not the problem. We are excluding it because once we plug it into the original equation we end up with logarithms of negative numbers. It is possible to have negative values of x be solutions to these problems, so don’t mistake the reason for excluding this value.

Also, along those lines we didn’t take

as a solution because it was positive, but because it didn’t produce any negative numbers or zero in the logarithms upon substitution. It is possible for negative numbers to not be solutions.

So, with all that out of the way, we’ve got a single solution to this equation,

In order to solve these kinds of equations we will need to remember the exponential form of the logarithm. Here it is if you don’t remember.

Example 2 Solve each of the following equations.

(a)

(b)

Solution

(a)

To solve these we need to get the equation into exactly the form that this one is in. We need a single log in the equation with a coefficient of one and a constant on the other side of the equal sign. Once we have the equation in this form we simply convert to exponential form.

So, let’s do that with this equation. The exponential form of this equation is,

Notice that this is an equation that we can easily solve.

Now, just as with the first set of examples we need to plug this back into the original equation and see if it will produce negative numbers or zeroes in the logarithms. If it does it can’t be a solution and if it doesn’t then it is a solution.

Only positive numbers in the logarithm and so

is in fact a solution.

(b)

In this case we’ve got two logarithms in the problem so we are going to have to combine them into a single logarithm as we did in the first set of examples. Doing this for this equation gives,

Now, that we’ve got the equation into the proper form we convert to exponential form. Recall as well that we’re dealing with the common logarithm here and so the base is 10.

Here is the exponential form of this equation.