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

Solution Method 1: Using a Common Base 
An exponential equation in which each side can be expressed in terms of
the same base can be solved using this property:
if b^{x} = b^{y}, then x = y (where b > 0 and b ≠1).
If the bases are the same, set the exponents equal. 

Solve for x: 
Answer 
1. 

Since the bases are the same, set the exponents equal to one another and solve for x:
3x  2 = 2x + 1
x = 3 
2. 

27 can be expressed as a power of 3:

3. 

Both 8 and 32 can be expressed as powers of 2:

4. 

Both 125 and 1/25 can be expressed as powers of 5:

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, but when working with e, use ln.) 
Remember:


Solution Method 2: Using logs 
To solve most exponential equations:
1. Isolate the exponential expression.
2. Take log or ln of both sides, to set up the inverse relationship between exponentials and logarithms.
3. Use this inverse relationship:
(where a > 0, a ≠1, and log_{a}a^{x} is defined).
4. Solve for the variable.


Things to remember
about logs:


Grab your calculator!

Solve for x,
to nearest thousandth: 
Answer 
1. 
5^{x} = 7
* Can you see how trying to get a common base for 5 and 7 would be an extremely difficult task? The log method will save us a good deal of aggravation on this problem.
Also notice that the solution can be found using either log or log_{5}. 
OR

• Take the log of both sides.
• Apply the log power rule.
• Solve for x.
• Estimate answer from calculator
• log base 5 can also be used as a solution method.
• notice how the log_{5} of 5^{x} is really composition of inverses and yields x.
• Notice the change of base formula used at the end for the calculator.


2. 
15 = 3^{2x+1} 

OR 


3. 
e^{x} = 43 
Since the natural log is the inverse of the natural exponential function, use ln to quickly solve this problem.
ln e^{x} = ln 43
x = ln 43 ≈ 3.761

4. 


• 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. 

5. 


• Isolate the exponential
• Take the log of both sides
• Apply the log property
• Divide by log 4
• Estimate using calculator 

6. 


• First, divide by the coefficient to isolate the exponential
• Take the log of both sides
• Apply the log rule
• Divide by log 2
• Estimate answer 

7. 


• Isolate the exponential
• Divide each side by the coefficient of 2
• Take ln of both sides
• Remember that ln x and e^{x} are inverse functions. 

8. 

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 for this problem.

• Remember that e^{x} • e^{x} = e^{2x}
• This problem is really
x^{2}  4x + 3 = 0 where x = e^{x}
• Factor
• Both solutions are answers. 


For help with exponential equations on
your calculator,
click here.


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