Solve the logarithmic equation log 2 6 - x 3 - ... - JAMB Mathematics 2023 Question

Let's solve the logarithmic equation:

\[ \log_2(6 - x) = 3 - \log_2(x) \]

First, combine the logarithmic terms on the right side:

\[ \log_2(6 - x) + \log_2(x) = 3 \]

Now, use the product rule of logarithms (\(\log_a(b) + \log_a(c) = \log_a(b \cdot c)\)):

\[ \log_2[(6 - x) \cdot x] = 3 \]

Next, simplify the expression inside the logarithm:

\[ \log_2(6x - x^2) = 3 \]

Now, rewrite the equation in exponential form:

\[ 2^3 = 6x - x^2 \]

\[ 8 = 6x - x^2 \]

Bring all terms to one side to form a quadratic equation:

\[ x^2 - 6x + 8 = 0 \]

Now, factor the quadratic:

\[ (x - 4)(x - 2) = 0 \]

Set each factor equal to zero:

\[ x - 4 = 0 \] or \[ x - 2 = 0 \]

So, the solutions are:

\[ x = 4 \] or \[ x = 2 \]

Therefore, the correct answer is \(x = 4\) or \(x = 2\)