Question

Log x(3-x)*log x(4-x)-log x(x^2-7x+12)+1=0

Answer 1

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Answer 2

To solve the equation:

$\log_x(3-x) \cdot \log_x(4-x) - \log_x(x^2 - 7x + 12) + 1 = 0$

Let's break it down step by step:

Step 1: Simplify the equation.

First, you can simplify the equation by using logarithm properties. Recall that $\log_a{b} - \log_a{c} = \log_a{\frac{b}{c}}$. Apply this property to your equation:

$\log_x\left(\frac{(3-x)(4-x)}{x^2 - 7x + 12}\right) + 1 = 0$

Step 2: Eliminate the constant 1.

Subtract 1 from both sides of the equation:

$\log_x\left(\frac{(3-x)(4-x)}{x^2 - 7x + 12}\right) = -1$

Step 3: Remove the logarithm.

Use the fact that $\log_a{b} = c$ is equivalent to $a^c = b$. Apply this property:

$\frac{(3-x)(4-x)}{x^2 - 7x + 12} = x^{-1}$

Step 4: Eliminate the negative exponent.

Recall that $x^{-1} = \frac{1}{x}$. Apply this to the equation:

$\frac{(3-x)(4-x)}{x^2 - 7x + 12} = \frac{1}{x}$

Step 5: Clear fractions.

To eliminate the fractions, you can multiply both sides of the equation by $x$ to get rid of the fraction on the right side:

$x(3-x)(4-x) = 1(x^2 - 7x + 12)$

Step 6: Expand and simplify.

Now, expand both sides of the equation:

$x(12 - 7x + x^2) = x^2 - 7x + 12$

Simplify further:

$12x - 7x^2 + x^3 = x^2 - 7x + 12$

Step 7: Rearrange the equation.

Move all terms to one side of the equation:

$x^3 - 7x^2 + 12x - x^2 + 7x - 12 = 0$

Simplify:

$x^3 - 8x^2 + 19x - 12 = 0$

Now you have a cubic equation. You can try to solve this equation by factoring, using the Rational Root Theorem, or numerical methods. The exact solutions might not be simple, so you may need to use a calculator or computer software to find the roots.