How do i get the bottom x away?
3=x^2-6x+8/x

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Solve the logarithmic equation

log₃(x – 4) + log₃(x – 2) – log₃ x = 1

Condition:  x > 4,  because logarithms are only defined for positive numbers.

log₃(x – 4) + log₃(x – 2) – log₃ x = 1

log₃(x – 4) + log₃(x – 2) – log₃ x = log₃ 3

Applying log properties, you can rewrite that equation as



Since log is an one-to-one function, you can "cancel out" those logs at both sides, so you have



Multiply both sides by  x  to simplify that denominator:



Multiply out those brackets, by applying the distributive property:



Subtract  3x  from both sides, and then combine like terms together:



Now you have a quadratic equation, where the coefficients are

a = 1,  b = – 9,  c = 8

Solve it using the quadratic formula:

Finding the discriminant  Δ:

Δ = b² – 4ac

Δ = (– 9)² – 4 · 1 · 8

Δ = 81 – 32

Δ = 49

Δ = 7²

Then,





You can discard  x = 1  as a solution, because those initial logarithms are not defined for this value of x (remember that  x  must be greater than 4).

So the only solution is  x = 8.

Solution set:  S = {8}.

I hope this helps. =)

Tags:  logarithmic logarithm log equation condition quadratic formula discriminant solve algebra