SOLVE EQUATIONS WITH VARIABLE EXPONENTS

The following steps will be useful to solve equations in which the variables are in exponent. 

Step 1 :

On both sides first we have to simplify using exponent rules.

Step 2 :

After rewriting exponential equations with same base on both sides, we can compare the powers. Because if two exponential terms are equal with same base, then their exponents also must be equal. 

Step 3 :

Equate the exponents and solve for the unknown / variable. 

Solved Examples

Example 1 :

Solve for m :

5^m ÷ 5^-3  =  5⁵

Solution :

5^m ÷ 5^-3  =  5⁵

5^{m - (-3)}  =  5⁵

5^{m + 3}  =  5⁵

Then, 

m + 3  =  5

Subtract 3 from each side.

m  =  2

Example 2 :

Solve for m :

4^m  =  64

Solution :

4^m  =  64

4^m  =  4³

Then, 

m  =  3

Example 3 :

Solve for m :

8^{m - 3}  =  1

Solution :

8^{m - 3}  =  1

8^{m - 3}  =  2⁰

Then, 

m - 3  =  0

Add 3 to each side. 

m  =  3

Example 4 :

Solve for m :

(a³)^m  =  a⁹

Solution :

(a³)^m  =  a⁹

a^3m  =  a⁹

Then, 

3m  =  9

Divide each side by 3. 

m  =  3

Example 5 :

Solve for m :

(5^m)² x 25³ x 125²  =  1

Solution :

(5^m)² x 25³ x 125²  =  1

5^2m x (5²)³ x (5³)²  =  1

5^2m x 5⁶ x 5⁶  =  1

5^{2m + 6 + 6}  =  1

5^{2m + 12}  =  1

5^{2m + 12}  =  5⁰

Then, 

2m + 12  =  0

Subtract 12 from each side.

2m  =  -12

Divide each side by 2. 

m  =  -6

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