Question

For 8^2x - 16.8^x + 48 = 0, the sum of values of x is equal to:

• A. 1 + log₆ 8
• B. 1 + log₈ 6
• C. log₈ 6
• D. 16

Answer 1

The Correct Option is B

To solve the given equation \(8^{2x} - 16 \cdot 8^x + 48 = 0\) and find the sum of the values of \(x\), let's start by making a substitution to simplify the equation.

Let \(y = 8^x\). Then, \(8^{2x} = (8^x)^2 = y^2\). 

Substituting these into the equation, we have:

\(y^2 - 16y + 48 = 0\)

This is a standard quadratic equation. We apply the quadratic formula to solve for \(y\):

The quadratic formula is given by \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -16\), and \(c = 48\).

Calculating the discriminant:

\(b^2 - 4ac = (-16)^2 - 4 \cdot 1 \cdot 48 = 256 - 192 = 64\)

Since the discriminant is a perfect square, the roots are real and distinct.

Substituting into the quadratic formula:

\(y = \frac{16 \pm \sqrt{64}}{2} = \frac{16 \pm 8}{2}\)

This gives us two potential roots for \(y\):

• \(y_1 = \frac{16 + 8}{2} = 12\)
• \(y_2 = \frac{16 - 8}{2} = 4\)

Recall that \(y = 8^x\), so:

For \(y_1 = 12\): \(8^x = 12\)

Taking the logarithm base 8 on both sides, we have:

\(x_1 = \log_8 12\)

For \(y_2 = 4\): \(8^x = 4\)

Similarly, taking the logarithm base 8 on both sides:

\(x_2 = \log_8 4\)\)

Calculating these logarithms:

• \(x_1 = \log_8 12 = \log_8 (2^2 \cdot 3) = \log_8 4 + \log_8 3\)
• \(x_2 = \log_8 4 = \log_8 2^2 = 2\log_8 2 = \frac{2}{3}\)

The sum of the values of \(x\) is:

\(x_1 + x_2 = \log_8 4 + \log_8 3 + \frac{2}{3} = \log_8 (4 \cdot 3) + \frac{2}{3} = \log_8 12 + \frac{2}{3}\)

After further simplification, this reduces to the expression that matches one of the given options.

Thus, the correct answer is: 1 + \(\log_8 6\).