A-Level Edexcel Maths Pure Differentiation: 7. (i) Find the exact value of x

7. (i) Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$ (ii) Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Question 9

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7. (i) Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$ (ii) Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a. Give y... show full transcript

Worked Solution & Example Answer:7. (i) Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$ (ii) Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Step 1

Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$

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Answer

To solve for x, start by applying the properties of logarithms:

Rewrite the equation: $\log_2(2x) = \log_2(5x + 4) - 3$
Move the logarithmic term to one side: $\log_2(2x) + 3 = \log_2(5x + 4)$
Rewrite 3 in terms of logarithms: $\log_2(2x) + \log_2(2^3) = \log_2(5x + 4)$
Combine the logarithms: $\log_2(2x \cdot 8) = \log_2(5x + 4)$ which simplifies to: $\log_2(16x) = \log_2(5x + 4)$
Set the arguments of the logarithms equal: $16x = 5x + 4$
Solve for x: $16x - 5x = 4$
$11x = 4$
$x = \frac{4}{11}$

Step 2

Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a.

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Answer

Simplify the logarithmic expression:
Since $\log_y y = 1$ , we can replace it: $1 + 3\log_2 2 = 5$
Recall that $\log_2 2 = 1$ : $1 + 3 \cdot 1 = 5$ simplifies to: $1 + 3 = 5$
This means we need to express y in terms of another variable a: To isolate y, we can use: $3 + 1 = 5$ implies y must follow $y = a^k$ format for some base a. Depending on the context of the problem, you can express y in the simplest terms as: $y = 2^{5-3} = 2^{2} = 4$ .

7. (i) Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$ (ii) Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Worked Solution & Example Answer:7. (i) Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$ (ii) Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a - Edexcel - A-Level Maths Pure - Question 9 - 2013 - Paper 4

Find the exact value of x for which $$ \log_2(2x) = \log_2(5x + 4) - 3 $$

Given that $$ \log_y y + 3\log_2 2 = 5 $$ express y in terms of a.

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