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 $y = 10^x$. Then, the equation becomes:
\[ y + \frac{4}{y} = \frac{81}{2} \]
Multiply through by $y$ to eliminate the fraction:
\[ y^2 + 4 = \frac{81y}{2} \]
Multiply through by 2 to clear the denominator:
\[ 2y^2 + 8 = 81y \]
Rearrange:
\[ 2y^2 - 81y + 8 = 0 \]
Now, solve this quadratic equation using the quadratic formula:
\[ y = \frac{-(-81) \pm \sqrt{(-81)^2 - 4(2)(8)}}{2(2)} \]
\[ y = \frac{81 \pm \sqrt{6561 - 64}}{4} = \frac{81 \pm \sqrt{6497}}{4} \]
Taking the roots, we find that $y = 10^x$, so:
\(\boxed{2\log_{10}2}\)
Therefore, the sum of all distinct real values of $x$ is $2\log_{10}2$.