Question

The sum of all distinct real values of x that satisfy the equation \(10^x + \frac{4}{10^x} = \frac{81}{2}\)  is

• A. \(3\log_{10} 2\)
• B. \(\log_{10} 2\)
• C. \(4\log_{10} 2\)
• D. \(2\log_{10} 2\)

Answer 1

The Correct Option is D

Let $y = 10^x$. The equation transforms to:
\[ y + \frac{4}{y} = \frac{81}{2} \]
To remove the fraction, multiply the entire equation by $y$:
\[ y^2 + 4 = \frac{81y}{2} \]
Next, clear the denominator by multiplying by 2:
\[ 2y^2 + 8 = 81y \]
Rearrange the terms to form a standard quadratic equation:
\[ 2y^2 - 81y + 8 = 0 \]
Use the quadratic formula to solve for $y$:
\[ y = \frac{-(-81) \pm \sqrt{(-81)^2 - 4(2)(8)}}{2(2)} \]
Simplify the expression:
\[ y = \frac{81 \pm \sqrt{6561 - 64}}{4} = \frac{81 \pm \sqrt{6497}}{4} \]
Since $y = 10^x$, the values for $x$ are the logarithms of these $y$ values. The distinct real values of $x$ lead to a sum of:
\(\boxed{2\log_{10}2}\)