We have methods for solving equations such as
. But what happens if we have an equation
where the variable
is in the exponent. If
, we cannot divide by 3 or subtract 3 from both sides of the equation
to solve for x. Or if the equation
we still have a problem isolating the variable. We need a process for solving exponential equations. Exponential equations
come in two forms.
- In one case, it is possible to get the same base on each side of the equation. When this happens, there is a rule that says if the bases are the same, then the exponents must be the same also.
If a > 0 and a 1, then ab = ac is true if and only if b = c.
Let’s first practice with equations of the first type.
- In the other type of exponential equation we are not able to get the same base on both sides of the equation and will have to have a different process for solving.
You may not know right away if it is possible to change 243 so that it is an exponential expression with a base of 3. You might want to find some values of the powers of 3 to see if it is possible. We know and can continue until we find what we need.
So now that we know we can write 243 as a power of 3, we re-write the equation so that both sides have a base of 3.
In some cases, further simplification is necessary at this point, but we do not have to worry in this case. The next example will demonstrate the need to simplify before moving on.
You can now set your exponents equal to each other:
x = 5 Since this equation is already solved for x, we do not have to do any more work. There will be occasions where solving the equation will require more steps.
You should always check your solution in the original equation.
Yes, 35 = 243, our solution is correct.
Our first question needs to be “Can I get 27 and 9 to have the same base?” The answer is yes. Both 27 and 9 can be written as powers of 3. ( and ) Our equation can now be written as
The bases are now both 3, but the exponents need to be simplified. We will need to use rules of exponents to do this simplification. If you need to refresh your memory on rules of exponents, click here.
Now that the bases are the same and the exponents are simplified you can set the exponents equal to each other:
NOTE: make sure that you remember to distribute the 2 to both the x and the 1!
Solve the new equation for x:
Be sure to check your answer.
When you use your calculator to evaluate each side you see that
and The solution checks and
The second type of exponential equation
cannot be written so that both sides have the same base. Recall the equation
we used at the beginning of the lesson
. No matter
how we try, there will never be a way to write 11 as a power of 3.
To solve this type of exponential equation, we must make use of logarithms. If you have not studied laws of logarithms or need a reminder, click here
. Let’s now practice solving the second type of equations by actually going through the process step by step.
The solution to this type of equation involves taking the logarithm of both sides. Since a logarithm must have a base, let’s choose to use base 3 since that is also the base of the exponential equation. This will give us
Initially this looks like a much uglier equation, but a logarithm law allows us to simplify this equation to
This is one form of an answer to the equation, but most of the time we will approximate the answer to this type of equation using the calculator. However, the calculator can only evaluate a logarithm with a base of 10 or a base of e. To evaluate this answer on the calculator, you will need to use the change of base formula. (link to logs-rules.doc) and then input the fraction into the calculator.
Make sure you check your answer which gives . Remember that we rounded our calculator answer to three decimal places, so when using that rounded value in checking our answer, there will be a round-off error. Do not panic. We haven’t done anything wrong. Our rounding just creates a less exact answer.
We normally begin by asking ourselves if we can get the same base on both sides. In this case, we can’t answer that question yet because we do not have just the exponential expression on one side of the equation. We need to divide both sides by 3 before beginning this problem. The equation is now
We can now see for sure that 2 and 7 cannot be written with the same base. We will use our process of taking the log of both sides. In this case, we will use a log of base 2.
Now one of our law of logs says we can simplify this equation to
To solve this equation for x, we need to divide both sides by 4 and get
Again, we would like to get a decimal approximation for this value. It might be easier to work with the and the change of base and then divide by 4.
Now take that approximation and divide by 4 to get 0.702. So
x = 0.702. Checking this answer will give 21.009 which is acceptable considering our rounding throughout the problem.