Question

`log_(2 )(x - 6) + log_(2)(x - 4) = log_(2)(x)`x=8,3x=8No solution

Answer 1

x=8,3

Step-by-step explanation:

Given the expression:

`\log _2\mleft(x-6\mright)+log_2\mleft(x-4\mright)=log_2\mleft(x\mright)`

Applying the addition law of logarithm:

`\log _2(x-6)(x-4)=log_2x`

Next, cancel the logarithm operator on both sides:

`\begin{gathered} (x-6)(x-4)=x \\ x^2-4x-6x+24=x \\ x^2-10x-x+24=0 \\ x^2-11x+24=0 \end{gathered}`

We solve the resulting quadratic equation:

`\begin{gathered} x^2-8x-3x+24=0 \\ x(x-8)-3(x-8)=0 \\ (x-3)(x-8)=0 \\ x-3=0\text{ or }x-8=0 \\ x=3\text{ or }x=8 \end{gathered}`

The value of x is 3 or 8.