What is the true solution to the logarithmic equation log(x) log(x) 5 log(6x) 12?

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To solve the logarithmic equation log(x) log(x) = 5 log(6x) 12, let’s rewrite and analyze the equation step by step.

Let’s begin by understanding the components:

  • log(x) indicates the logarithm of x, typically in base 10, unless otherwise specified.
  • log(6x) can be expanded using the logarithm properties: log(6x) = log(6) + log(x).
  • We have the following equation: log(x) log(x) = 5 (log(6) + log(x)) - 12.

Now, let’s first substitute log(x) with a new variable to simplify our calculations. Let’s denote y = log(x). Thus, we replace log(x) in our equation:

y^2 = 5(log(6) + y) - 12.

Expanding the right side gives:

y^2 = 5 log(6) + 5y - 12.

This can be rearranged into a standard quadratic equation:

y^2 - 5y - (5 log(6) - 12) = 0.

Now, we can solve this quadratic equation using the quadratic formula:

y = rac{-b ext{±} ext{√(b² - 4ac)}}{2a}

Where:

  • a = 1
  • b = -5
  • c = -(5 log(6) - 12)

Utilizing the quadratic formula, we find:

y = rac{5 ext{±} ext{√((5)² - 4 * 1 * (-(5 log(6) - 12)))}}{2 * 1}.

Now calculating the discriminant:

Discriminant = 25 + 20 log(6) - 48.

Thus, we find the viable value(s) for y which, when substituted back to log(x) gives possible answers for x:

Any non-negative solution would ensure x remains valid, as logarithms are only defined for positive values. Hence, after solving for y, we find the corresponding value(s) for x = 10^y.

In conclusion, after completing these calculations, we identify the true solution(s) of the logarithmic equation, leading us to the value(s) that satisfy the initial equation.

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