Exponential and Logarithmic Equations

 When you are done with this section, you will be able to do the following

There are two basic methods that you con use to solve exponential and logarithmic equations.

One method uses the definition of the function.  Since exponentials and logarithms are functions, if you have something like , then you know that the exponents must be equal.  This leaves us to solve the equation 5x - 3 = 10. 

This same process is true of logarithmic equations which are of a similar form to .  This leaves us to solve the equation 3x + 5 = 8x.  When solving logarithmic equations, we must always check our answer to ensure that the value or values of x that we get are in the domain of our logarithmic function.  In this case, we get that x = 1.  The domain of the original logarithmic functions are .  x = 1 is in the domain of the functions.

This method becomes difficult when the exponential equation has two exponentials with different bases or when the logarithmic functions are of different bases.  This is where the second method of solving equations comes in.  In this method, we use the concept that the logarithmic and exponential functions are inverses of each other.  (Remember the inverse functions "undo" each other.)

There are two approaches to this method.

Approach 1:  This approach is my preferred approach.  If you are solving an exponential equation, you want to change it to logarithmic form.  If you are solving a logarithmic equation, you want to change it to exponential form.
Example 1:  Solve the following equation for x
Solution: 
The first step to solving this is to change the equation to logarithmic form.

(Remember back to the lesson on logarithmic functions where I said "You will need to be able to easily switch from exponential form to logarithmic form and back.".  This is the time for that.)

In this problem b is 23, a is 2, and c is 5x - 3.  When we put this into logarithmic form we get .  We can now go through the process of solving for x.  This will give us


This is an exact answer for this problem.  If we need a decimal answer, we can use the change of base formula to calculate that.  This would give us .
Example 2:  Solve the following equation for x
Solution: 
The first step to solving this is to change the equation to logarithmic form.

(Remember back to the lesson on logarithmic functions where I said "You will need to be able to easily switch from exponential form to logarithmic form and back.".  This is the time for that.)

In this problem c is 8, a is 4, and b is 3x +5.  When we put this into exponential form we get .  We can now go through the process of solving for x.  This will give us


This is an exact answer for this problem. 

Approach 2:  If you are solving an exponential equation, you "take the logarithm" of both sides of the equations.  If you are solving a logarithmic equation, you "exponentiate" both sides of the equation.
Example 3:  Solve the following equation for x
Solution: 
The first step to solving this is to "take the log" of both sides of the equation.  We can use any base for the logarithm.  The standard bases to use are base 10 and base e since these bases are on our calculator.  It is common to use the base that will make the next steps easiest..

We will use log (i.e.:  logarithm base 10).  This will give us .  Next we need to use one of the "Laws" of logarithms to move the exponent to be a multiplier to get .  We can now go through the process of solving for x.  This will give us


This is an exact answer for this problem. 

You should try going through this problem by taking the natural logarithm (ln) of both sides.  You should see that you get the same answer.

Example 4:  Solve the following equation for x
Solution: 
The first step to solving this is to "exponentiate" both sides of the equation.  Since the base of the logarithm is 4, we will use 4 as the base of our exponentiation.

Exponentiating both sides will give us .  Using one of the properties of logarithms in the logarithmic functions section, we can simplify the left has side of the equation to get (property of logarithms used: ).  We can now go through the process of solving for x.  This will give us


This is an exact answer for this problem. 

All of these examples have had the same basic form of (a number raised to a power equals a number) or (a single logarithm of something equals a number).  Any problem that you do must first be manipulated to get into one of these forms before you go through either of these approaches.  This means that the you may need to "isolate" the exponential or logarithm.  To do this you may need to so some addition/subtraction and multiplication/division.  It may also be necessary to combine multiple logarithmic expressions into a single logarithmic expression using the "Laws" of logarithms.

A good place to find information about exponential and logarithmic equations is your text book (pages 412-418) in College Algebra 3rd edition by Stewart, Redlin, and Watson).


Additional Exponential Equations On-line Resources:
Additional Logarithmic Equations On-line Resources:

Exponential and Logarithmic Equations Homework:
The Exponential and Logarithmic Equations Homework contains 7 problems.
Things to remember:
  • When putting in answers, the only logarithms that you can use are log (log base 10) and ln (log base e).  If you have a logarithm with any other base, you will need to use the change of base formula.
  • If you put in your answers in decimal form, you must have at least 4 digits to the right of the decimal.
  • In solving this type of equation, do not round any number until the very end of the problem.
Homework examples:
also-recommended Homework examples:
  • Example 6:  similar to problem 6 also-recommended
  • Example 7:  similar to problem 3 also-recommended
  • For problem 2 also-recommended, you should solve it like a quadratic equation.  Here the x is replaced with ex.

    •
Keep in mind that it is recommended that you also complete all the problems in the set called "also recommended" with the same number as the Exponential and Logarithmic Equations Homework.  There are many different ways to ask the same questions.  This will allow you to see additional problems that are related to this topic.

© 2005 Elizabeth E. K. Jones and the ASU Department of Mathematics and Statistics - All rights reserved.