Use the product rule for logarithms to find all x values ...

Use the product rule for logarithms to find all x values such that log12 2x 6 log12 x 2 2.

Solution First, use the product property of logarithms to condense the left-hand-side to a single logarithm

log12 2x 6 log12 x 2 2

log12 2x 6 x 2 2

Convert the equation from logarithmic form to exponential form.

log12 2x 6 x 2 2

122 2x 6 x 2 144 2x 6 x 2

Solve the resulting quadratic equation.

144 2x 6 x 2

144 2x2 10x 12 0 2x2 10x 132

0 2 x2 5x 66

0 2 x 11 x 6

x 11 x6

The solution of x 11 is not in the domain of the function, making x 6 the only solution.

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