Consider $8^{x-4}=8^{10}$. Because the bases are equal, the exponents must also be equal. Complete the solution to the equation is $x=$?

Since the bases are equal, equate the exponents: x − 4 = 10 .
Add 4 to both sides of the equation to solve for x : x = 10 + 4 .
Calculate the value of x .
The solution to the equation is 14 ​ .

Explanation

Equating the exponents We are given the equation 8 x − 4 = 8 10 . Since the bases are equal, the exponents must also be equal. This gives us the equation x − 4 = 10 .

Solving for x To solve for x , we add 4 to both sides of the equation: x − 4 + 4 = 10 + 4 x = 14

Final Answer Therefore, the solution to the equation is x = 14 .


Examples
Imagine you are comparing the growth of two investments. If both investments grow at the same rate (the base) and you want to find out when their total growth will be equal, you can use the principle of equating exponents. For example, if Investment A grows as 2 t − 1 and Investment B grows as 2 5 , where t is time in years, setting the exponents equal ( t − 1 = 5 ) helps you determine when the investments will have the same value. This concept is useful in finance, biology (population growth), and computer science (algorithm complexity).

Answered by GinnyAnswer | 2025-07-03

The solution to the equation 8 x − 4 = 8 10 is found by setting the exponents equal, leading to x = 14 .
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Answered by Anonymous | 2025-07-04