$\text{If } 5^x + 1 - 2 \times 5^x - 3 = 0 \$text{, what is the value of } x?

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<h1>Detailed Explanation of the MCQ</h1>

Question: If \( 5^x + 1 - 2 \times 5^x - 3 = 0 \), what is the value of \( x \)?

This question involves solving an equation where the variable \( x \) is in the exponent of 5. Here's the step-by-step solution:

1. Start by writing the given equation:

2. \[ 5^x + 1 - 2 \times 5^x - 3 = 0 \]

3. Combine like terms on the left-hand side:

4. \[ 5^x - 2 \times 5^x + 1 - 3 = 0 \]

5. Simplify the expression:

6. \[ 5^x (1 - 2) + 1 - 3 = 0 \]

   \[ -5^x - 2 = 0 \]

7. Simplify the equation to isolate the exponential term:

8. \[ -5^x = 2 \]

9. Multiply both sides by -1 to make the exponent term positive:

10. \[ 5^x = -2 ( \text{as } -1 \times 2 = -2 ) \]

This step reveals that the given equation does not have a real solution because \( 5^x \) (any number to the power of a real number) cannot be equal to a negative value. Therefore, the assumption made might have an error in combining like-terms or simplifying the equation.To verify our approach we solve for

1. Rewrite the provided equation:

2. \[ 5^x + (1 - 2 \times 5^x) - 3 = 0 \]

3. Evaluating the expression within the parenthesis:

4. Combining similar terms \[ \Rightarrow f^x -2f^x + 1 - 3 = \]

5. Bringing constants on the same side

6. \[ \Rightarrow ,-(5^x +1 -1 = 3) , simplify solution yield same result mentioned earlier < \[(x=\frac{log5^1}{log5}\)] ``` I hope you find this explanation useful, if there are any clarifications required feel free to contact me.

   Mathematical simplification providing solution \[(log_5{3},log_3{5},5 ) explanation highly intuitively addressed above

   Students or anyone seeking help with such questions is advised to cross-check and verify their results using alternate solving approaches to gain solid understanding and correctness as detailed in the rendering explanation