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Which Is the Only Solution to the Equation LOG3(X^2 + 6X) = LOG3(2X + 12)?

In mathematics, equations play a crucial role in solving problems and understanding the relationships between variables. One such equation that requires careful analysis is LOG3(X^2 + 6X) = LOG3(2X + 12). In this article, we will explore the steps to find the only solution to this equation and provide some frequently asked questions related to logarithmic equations.

To solve the equation LOG3(X^2 + 6X) = LOG3(2X + 12), we need to understand the properties of logarithms and apply them appropriately. The equation involves the base 3 logarithm, which means that the logarithm of a number is taken with respect to the base 3. Let’s break down the process step by step:

Step 1: Apply the property of logarithms

The property of logarithms states that if LOGa(b) = LOGa(c), then b = c. In our equation, we have LOG3(X^2 + 6X) = LOG3(2X + 12). Applying the logarithmic property, we can write X^2 + 6X = 2X + 12.

Step 2: Simplify the equation

To solve for X, we need to simplify the equation obtained from step 1. Rearranging the terms, we have X^2 + 4X – 12 = 0.

Step 3: Factor or use the quadratic formula

To find the solutions to a quadratic equation, we can either factor it or use the quadratic formula. Factoring the equation X^2 + 4X – 12 = 0, we get (X – 2)(X + 6) = 0. Therefore, X = 2 or X = -6.

Step 4: Check for extraneous solutions

Now that we have obtained two possible solutions, we need to verify if they satisfy the original equation. Plugging X = 2 into the original equation, we get LOG3(2^2 + 6(2)) = LOG3(2(2) + 12), which simplifies to LOG3(16) = LOG3(16). Both sides of the equation are equal, so X = 2 is a valid solution.

Similarly, plugging X = -6 into the original equation, we get LOG3((-6)^2 + 6(-6)) = LOG3(2(-6) + 12), which simplifies to LOG3(0) = LOG3(0). However, the logarithm of 0 is undefined, so X = -6 is not a valid solution.

Therefore, the only solution to the equation LOG3(X^2 + 6X) = LOG3(2X + 12) is X = 2.

FAQs:

Q1: What are logarithms?

A1: Logarithms are mathematical functions that represent the exponent to which a base number must be raised to obtain a given number. In the equation LOG3(X^2 + 6X) = LOG3(2X + 12), the base is 3.

Q2: What is the property of logarithms used in this equation?

A2: The property of logarithms used in this equation is the equality property, which states that if LOGa(b) = LOGa(c), then b = c.

Q3: Why is X = -6 not a valid solution?

A3: X = -6 is not a valid solution because plugging it into the original equation leads to LOG3(0) = LOG3(0), which is undefined. Logarithms of 0 are not defined in mathematics.

Q4: Are there any other methods to solve this equation?

A4: Yes, instead of factoring, you can also use the quadratic formula to solve for X. The quadratic formula is X = (-b ± √(b^2 – 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q5: Can logarithmic equations have multiple solutions?

A5: Yes, logarithmic equations can have multiple solutions. However, in this particular equation, there is only one valid solution, which is X = 2.

In conclusion, the only solution to the equation LOG3(X^2 + 6X) = LOG3(2X + 12) is X = 2. By understanding the properties of logarithms and following the steps outlined above, we can successfully solve this equation.

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