Exponential Equations
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An exponential equation is one in which a variable occurs in the exponent.

An exponential equation in which each side can be expressed in
terms of the same base can be solved using the property:
 
  If the bases are the same, set the exponents equal.

Examples:

 

  Solve for x.

  Answer

1.    Since the bases are the same, set the exponents equal to one another:
2x + 1 = 3x - 2
3 = x
2.    27 can be expressed as a power of 3:

2x - 1 = 3x
-1 = x
3.    25 can be expressed as a power of 5:

3x - 8 = 4x
-8 = x


If you can express both sides of the equation as powers of the same base, you can set the exponents equal to solve for x.

 

Unfortunately, not all exponential equations can be expressed in terms of a common base.  For these equations, logarithms are used to arrive at a solution.  (You may solve using common log or natural ln.)

Remember:

To solve most exponential equations:
1.  Isolate the exponential expression.
2.  Take log or ln of both sides.
3.  Solve for the variable.
 
Things to remember about logs:

 


 

Solve for x,
to the nearest thousandth.

  Answer  
1.   
• Take the log of both sides.
• Apply the log rule for exponents shown above.
• Solve for x.
• Estimate answer using calculator.
OR
• log base 5 can also be used as a solution method
• notice how the log5 of 5x is really composition and yields x.
• notice the change of base formula used at end
2.   
OR
Also, log base 3 can be used.
3.   
Also, log base 1/2 can be used as a solution method.
 
4.  
OR
5.   Since the natural log is the inverse of the exponential function, use ln to quickly solve this problem.

 
6.  
• First, get rid of the coefficient of the exponential term (divide by 150).

• Now, proceed using ln to quickly solve.

• Do not round too quickly.  Be sure to carry enough decimal values to allow you to round to thousandths (in this case) for the final answer.
7.   
• Isolate the exponential

• Take the log of both sides

• Apply the log property

• Divide by log 4

• Estimate using calculator
8.  
•  Isolate the exponential

•  Divide each side by the coefficient of 2
• Take ln of both sides
• Remember that ln x and ex are inverse functions.
9.
• First, divide by the coefficient to isolate the exponential

•  Proceed as shown above.
10. This question requires some additional thinking.  Because of the differing powers of e, our previous methods will not be of much help.  We will need a different strategy with this problem.
• Remember that ex • ex = e2x

• This problem is really
x2 - 4x + 3 = 0 where x = ex.

• Factor

• Both solutions are answers.

 

How to use your
TI-83+/84+ graphing calculator with exponential equations.
Click calculator.