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Using the laws of logarithms, solve the equation $$\log_3 (12y + 5) - \log_3 (1 - 3y) = 2$$ - Edexcel - A-Level Maths Pure - Question 4 - 2021 - Paper 1

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Using the laws of logarithms, solve the equation $$\log_3 (12y + 5) - \log_3 (1 - 3y) = 2$$

Worked Solution & Example Answer:Using the laws of logarithms, solve the equation $$\log_3 (12y + 5) - \log_3 (1 - 3y) = 2$$ - Edexcel - A-Level Maths Pure - Question 4 - 2021 - Paper 1

Step 1

Step 1: Apply the Laws of Logarithms

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Answer

Using the property of logarithms: log⁡ab−log⁡ac=log⁡a(bc)\log_a b - \log_a c = \log_a \left( \frac{b}{c} \right) we can rewrite the equation as: log⁡3(12y+51−3y)=2\log_3 \left( \frac{12y + 5}{1 - 3y} \right) = 2

Step 2

Step 2: Exponential Form

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Answer

Next, we convert the logarithmic equation to its exponential form: 12y+51−3y=32\frac{12y + 5}{1 - 3y} = 3^2 This simplifies to: 12y+51−3y=9\frac{12y + 5}{1 - 3y} = 9

Step 3

Step 3: Cross Multiply

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Answer

Cross-multiplying gives: 12y+5=9(1−3y)12y + 5 = 9(1 - 3y) Expanding the right-hand side results in: 12y+5=9−27y12y + 5 = 9 - 27y

Step 4

Step 4: Bring Like Terms Together

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Answer

Now, rearranging the equation we get: 12y+27y=9−512y + 27y = 9 - 5 This simplifies to: 39y=439y = 4

Step 5

Step 5: Solve for y

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Answer

Finally, divide both sides by 39: y=439y = \frac{4}{39}

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