Solve Exponential Equations

Examples of exponential equations

• 2^x = 4
• 2^2x = 16
• 2^{x + 1} = 256

Steps to solve exponential equations

There are many different kinds of exponential equations. First, let's focus on exponential equations that have a single term on both sides. These equations can be classified into two different types.

Type #1) When the bases are of both terms are the same

Type #2) When the bases are of the terms are different

Solving Exponential Equations of the same base (type #1)

Example: 4^x+1 = 4⁹

Step 1)Ignore the bases, and simply set the exponents equal to each other	1) X+ 1 = 9
Step 2) Solve for the variable	2) X = 9 − 1   X = 8

Solving Exponential Equations of with unlike bases (type #2)

Example Equation: 4³ =2^x

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "4 and 2 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 2 4 = 2²
2) Substitute the rewritten bases into original equation	2) 43 =2x (22)3 =2x
3) Simplify exponents	3) 26 = 2x
4) Solve like an exponential equation of like bases	4) 6 = x

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Demonstration Problem 2: Solving Exponential Equations

Exponential Equations with Negative Exponents

It is possible to use the steps outlined on this page to solve exponential equations with negative exponents. Again, it is necessary that both of the bases are powers of a common base. See the example below

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Problem 1) Solve the following exponential Equation:

9^x = 81

Answer

This is the second type of equation (exponents with unlike bases). So follow the steps.

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "9 and 81 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 3 and of 9 . You can use either base. I will use 3. 81 = 34 9 = 32
2) Substitute the rewritten bases into original equation	2)(32)x =34
3) Simplify exponents	3)32x =34
4) Solve like an exponential equation of like bases	4) 2x = 4 x = 2

Problem 2) Solve the exponential Equation below:

Answer

Since these equations have different bases, follow the steps for unlike bases.

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "1/4 and 32 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 2 1/4 = 2-2 32 = 25
2) Substitute the rewritten bases into original equation	2)(2-2)x =25
3) Simplify exponents	3)2-2x=25
4) Solve like an exponential equation of like bases	4) -2x = 5 x = -2/5

Problem 3) Solve the equation below:

4^2x+1 = 65

hint

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

Answer

Since these equations have different bases, followthe steps for unlike bases.

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "4 and 64 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 2 and of 4. You could use either base to solve this. I will use base 4 4 = 41 64 = 43
2) Substitute the rewritten bases into original equation	2)42x =43
3) Simplify exponents	3)42x =43
4) Solve like an exponential equation of like bases	4) 2x = 3 x = 3/2

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Problem 3) Solve this exponential equation:

hint

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

Answer

Since these equations have different bases, followthe steps for unlike bases.

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "1/9 and 27 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 3. 1/9 = 3-2 27 = 33
2) Substitute into original equation	2) (3-2)x=33
3) Simplify exponents	3)3-2x=33
4) Solve like an exponential equation of like bases	4) -2x = 3 x = -3/2

Problem 4) Solve this exponential equation:

hint

Rewrite this equation so that it looks like the other ones we solved--In other words, isolate the exponential expression as follows:

Answer

Since these equations have different bases, followthe steps for unlike bases.

1) Rewrite the bases as powers of a common base (Ignore the exponents and Answer the following question: "1/25 and 125 are powers of what number? " Once you have your answer, rewrite both of the bases as powers of that number)	1) They are both powers of 5. 1/25 = 5-2 125 = 53
2) Substitute into original equation	2)
3) Simplify exponents	3)5(-6x+8)=53
4) Solve like an exponential equation of like bases	4)