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2 log(x + a) = log(16a^r), where a is a positive constant Find x in terms of a, giving your answer in its simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 2

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2 log(x + a) = log(16a^r), where a is a positive constant Find x in terms of a, giving your answer in its simplest form. log(9y + b) - log(2y - b) = 2, where b is a... show full transcript

Worked Solution & Example Answer:2 log(x + a) = log(16a^r), where a is a positive constant Find x in terms of a, giving your answer in its simplest form - Edexcel - A-Level Maths Pure - Question 9 - 2016 - Paper 2

Step 1

Find x in terms of a

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Answer

To solve the equation, start by applying the power rule of logarithms:

2imeslog(x+a)=log(16ar)2 imes log(x + a) = log(16a^r)

This can be rewritten using the power rule:

log((x+a)2)=log(16ar)log((x + a)^2) = log(16a^r)

Since the logarithms are equal, we can equate the arguments:

(x+a)2=16ar(x + a)^2 = 16a^r

Taking the square root of both sides gives:

x+a=4ar/2x + a = 4a^{r/2}

Now, isolate x:

x=4ar/2−ax = 4a^{r/2} - a

Step 2

Find y in terms of b

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Answer

Starting with the equation:

log(9y+b)−log(2y−b)=2log(9y + b) - log(2y - b) = 2

Apply the quotient rule of logarithms:

log(9y+b2y−b)=2log\left(\frac{9y + b}{2y - b}\right) = 2

Exponentiate both sides to remove the logarithm:

This simplifies to:

Now, distribute the 100:

Rearranging the equation yields:

Which simplifies to:

Thus,

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