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Find the values of x such that $$2 \log_x{(x-2)} = 2$$ - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 3

Question 4

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Find the values of x such that $$2 \log_x{(x-2)} = 2$$

Worked Solution & Example Answer:Find the values of x such that $$2 \log_x{(x-2)} = 2$$ - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 3

Step 1

Rearranging the Equation

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Answer

First, we can rewrite the equation as:

$2 \log_x{(x-2)} = 2$

To simplify, divide both sides by 2:

$\log_x{(x-2)} = 1$

Step 2

Using the Definition of Logarithms

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Answer

Using the definition of logarithms, we convert the log equation:

$x^1 = x - 2$

This simplifies to:

$x = x - 2$

Step 3

Setting Up the Quadratic Equation

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Answer

We rearrange this to form:

$x - x + 2 = 0$

This leads us to:

$x^2 - 9x + 18 = 0$

Step 4

Factoring the Quadratic

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Answer

Next, we factor the quadratic:

$(x - 3)(x - 6) = 0$

Step 5

Finding the Solutions

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Answer

Setting each factor to zero gives us:

$x - 3 = 0 \quad \text{or} \quad x - 6 = 0$ Therefore, the solutions are:

$x = 3 \quad \text{and} \quad x = 6$

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