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May 5, 2016

Algebra - Notes
 Polynomial Functions Previous Chapter Next Chapter Systems of Equations Logarithm Functions Previous Section Next Section Solving Logarithm Equations

## Solving Exponential Equations

Now that we’ve seen the definitions of exponential and logarithm functions we need to start thinking about how to solve equations involving them.  In this section we will look at solving exponential equations and we will look at solving logarithm equations in the next section.

There are two methods for solving exponential equations.  One method is fairly simple, but requires a very special form of the exponential equation.  The other will work on more complicated exponential equations, but can be a little messy at times.

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

Note that this fact does require that the base in both exponentials to be the same.  If it isn’t then this fact will do us no good.

Let’s take a look at a couple of examples.

 Example 1  Solve each of the following. (a)     [Solution] (b)     [Solution] (c)     [Solution] (d)     [Solution] 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.                                                                  So, if we were to plug  into the equation then we would get the same number on both sides of the equal sign.   (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.                                                          And there is the answer to this part.

Now, the equations in the previous set of examples all relied upon the fact that we were able to get the same base on both exponentials, but that just isn’t always possible.  Consider the following equation.

This is a fairly simple equation however the method we used in the previous examples just won’t work because we don’t know how to write 9 as a power of 7.  In fact, if you think about it that is exactly what this equation is asking us to find.

So, the method we used in the first set of examples won’t work.  The problem here is that the x is in the exponent.  Because of that all our knowledge about solving equations won’t do us any good.  We need a way to get the x out of the exponent and luckily for us we have a way to do that.  Recall the following logarithm property from the last section.

Note that to avoid confusion with x’s we replaced the x in this property with an a.  The important part of this property is that we can take an exponent and move it into the front of the term.

we could use this property as follows.

The x in now out of the exponent!  Of course we are now stuck with a logarithm in the problem and not only that but we haven’t specified the base of the logarithm.

The reality is that we can use any logarithm to do this so we should pick one that we can deal with.  This usually means that we’ll work with the common logarithm or the natural logarithm.

So, let’s work a set of examples to see how we actually use this idea to solve these equations.