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LOGARITHMIC EQUATIONS
Definition

Any equation in the variable x that contains a logarithm is called a logarithmic equation. 
 

Recall the definition of a logarithm. This definition will be important to understand in order to be able to solve logarithmic equations. 

 
Examples

EXAMPLES OF LOGARITHMIC EQUATIONS
       
 

 

Example 1

  

 

Example 2

 
 

Log2 x = -5

5 + ln 2x = 4

 

 

Example 3

  

 

Example 4

 
 

ln x + ln (x - 2) = 1

log6 x + log6 (x + 1) = 1
Solving

STEPS TO SOLVE A logarithmic EQUATIONS

Your goal is to be able to use the definition of a logarithm. To use this, isolate the logarithmic expression to one side of the equation. All constants should be combined to the other side. Use properties of logarithms, if necessary, to combine logarithms to a single logarithmic term. Apply the definition - change to exponential form. Simplify the result. That's it!

Sample Problems

Sample Problem 1
 


 Log2 x = -5

2-5 = x 

   Answer: 
  1    = x
32       
This equation contains a single logarithmic expression on one side and a constant on the other side. Simply apply the definition of a logarithm. (i.e. change to exponential form.)

Sample Problems

Sample Problem 2
 


5 + ln 2x = 4
-5             -5 

ln 2x = -1  

e-1 = 2x

x = e-1 / 2

Answer: x » 0.1839
  1. Isolate the log term.
  1. Apply the definition of a logarithm. 
    (change to exponential form)
    Recall, "ln" is a logarithm whose base is the number e. 

 

 

 

 

 

  1. The answer can be approximated using a scientific calculator.

Sample Problems

Sample Problem 3
 

ln x + ln (x - 2) = 0
ln x(x - 2) = 0 

e0 = x(x - 2) 

1 = x2 - 2x
  x2 - 2x - 1 =0

 
    REJECT       ACCEPTABLE

Answer: 2.41

  1. Combine the two logarithms into a single logarithm. Recall:


 

 

  1. Change to exponential form using the definition of a logarithm.
  2. Solve the resulting equation. Here we have a quadratic equation. Since it is not factorable, we will solve using the quadratic formula.

 

  1. Since we cannot take the logarithm of zero or negative numbers. Reject any answers that will result in either of these in the original. 

Sample Problems

Sample Problem 4
 

log6 x + log6 (x + 1) = 1

log6 x(x + 1) = 1

61 = x(x + 1) 

x2 + x = 6

x2 + x - 6 = 0

(x + 3)(x - 2) = 0

x = -3   OR   x = 2
    REJECT       ACCEPTABLE

Answer: X = 2

  1. Combine the two logarithms into a single logarithm. Recall:
  2. Change to exponential form using the definition of a logarithm.
  3. Solve the resulting equation. Here we have a quadratic equation. Since it is factorable, we will solve using factoring.
  4. Since we cannot take the logarithm of zero or negative numbers. Reject any answers that will result in either of these in the original.