Solving the Equation: 2logxlog3log 5x2

To solve the equation 2logxlog3 log 5x2, we need to apply the properties of logarithms and solve the resulting quadratic equation. Let's go through the steps in detail.

Step-by-Step Solution

1. Initial Equation Reformulation

The equation is:

2log xlog 3 log 5x^2

We can start by simplifying the left-hand side:

2log xlog 3 log x^2log 3 log 3x^2

So the equation now looks like:

log 3x^2 log 5x^2

2. Equating the Arguments

Since the logarithms on both sides of the equation are equal, their arguments must also be equal:

3x^2 5x^2

Subtracting (5x^2) from both sides gives:

0 2x^2 - 5x - 2

Simplifying, we get a quadratic equation:

3x^2 - 5x - 2 0

3. Solving the Quadratic Equation

Using the quadratic formula (x frac{-b pm sqrt{b^2 - 4ac}}{2a}), where (a 3), (b -5), and (c -2), we get:

x frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 3 cdot (-2)}}{2 cdot 3}

x frac{5 pm sqrt{25 24}}{6}

x frac{5 pm sqrt{49}}{6}

x frac{5 pm 7}{6}

This gives us two solutions:

x frac{12}{6} 2

x frac{-2}{6} -frac{1}{3}

4. Checking the Solutions

The logarithm function is only defined for positive values, so we discard the negative solution (-frac{1}{3}). The valid solution is:

x 2

5. Verification

Substituting (x 2) back into the original equation:

2log 2log 3 log 5(2)^2

2log 2log 3 log 20

This holds true, confirming that (x 2) is the correct solution.

Understanding the Logarithm Properties

1. Logarithm Properties

For the equation 2logxlog3 log 5x2, we can use the following properties:

blog a log a^b

log alog b log ab

These properties help us simplify the equation before solving it.

Conclusion

By carefully applying the properties of logarithms and solving the resulting quadratic equation, we found that the valid solution to the equation 2logxlog3 log 5x2 is: