[tex]\frac{25^x-5^{2 x-1}}{4 \times 25^x}=\frac{1}{5}[/tex]
Answer (2)
Rewrite the terms using exponent rules: 2 5 x = 5 2 x and 5 2 x − 1 = × 5 2 x /5 .
Substitute back into the original equation and factor out 5 2 x from the numerator.
Cancel the common term 5 2 x from the numerator and denominator.
Simplify the equation to find that 5 1 = 5 1 , which is an identity, meaning the equation is true for all x .
The solution holds for all real numbers, so the equation is an identity: all real numbers
Explanation
Problem Analysis We are given the equation 4 × 2 5 x 2 5 x − 5 2 x − 1 = 5 1 and we need to solve for x .
Rewriting the equation First, we rewrite 2 5 x as ( 5 2 ) x = 5 2 x . Also, we rewrite 5 2 x − 1 as 5 5 2 x . Substituting these into the equation, we get 4 × 5 2 x 5 2 x − 5 5 2 x = 5 1 .
Factoring Next, we factor out 5 2 x from the numerator: 4 × 5 2 x 5 2 x ( 1 − 5 1 ) = 5 1 .
Canceling Now, we cancel out 5 2 x from the numerator and denominator, provided that 5 2 x = 0 , which is always true for any real number x :
4 1 − 5 1 = 5 1 .
Simplifying Simplifying the left side, we have 4 5 4 = 5 1 , which simplifies to 5 1 = 5 1 .
Conclusion Since the equation simplifies to an identity 5 1 = 5 1 , it is true for all values of x . This means that the original equation holds for all real numbers x .
Examples
Exponential equations like this one are used in various fields, such as calculating population growth, radioactive decay, and compound interest. For example, if you invest money in a bank account with compound interest, the amount of money you have after a certain time can be modeled using an exponential equation. Solving such equations helps you determine how long it will take for your investment to reach a certain value.
By rewriting the equation using exponent rules, we simplify the expression to find that it holds true for all values of x. This results in the equation being an identity. Therefore, the solution is all real numbers.
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