Pure: Algebra & functions

The domain of a rational function excludes values that make the denominator zero. Since x + 2 equals 0 when x is -2, this value must be excluded to prevent division by zero. Option A is incorrect because x = 3 merely makes the numerator zero, which is mathematically defined.

Using logarithm laws, log2(8) is 3 and log2(4) is 2, which sum to 5. Alternatively, log2(8) + log2(4) equals log2(32), which is 5. Option D is incorrect because 6 would result from multiplying the individual logarithms together instead of adding them.

Taking the natural logarithm of both sides of the equation gives 2x = ln(7), and dividing by 2 isolates x to yield ln(7)/2. A diagram for this topic isn't available yet — the worked example below walks the same reasoning step by step.

• !Option A is incorrect because it neglects to divide by the coefficient of 2 from the exponent.
• !Option D is incorrect because 6 would result from multiplying the individual logarithms together instead of adding them.
• !Option A is incorrect because x = 3 merely makes the numerator zero, which is mathematically defined.
• !Option C is incorrect because it uses the wrong signs in the factorization, which would instead correspond to the equation x^2 + 4x - 5 = 0.

Answer each from memory — you'll see instantly whether you're right and why.

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