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Given that $ ext{log}_g y = 2 ext{log}_g 7 + ext{log}_g 4 + rac{1}{2}$, find $y$ in terms of $a$ - AQA - A-Level Maths: Pure - Question 7 - 2018 - Paper 3
Question 7
Given that $ ext{log}_g y = 2 ext{log}_g 7 + ext{log}_g 4 + rac{1}{2}$, find $y$ in terms of $a$. When asked to solve the equation $2 ext{log}_g x = ext{log}_9 ... show full transcript
Worked Solution & Example Answer:Given that $ ext{log}_g y = 2 ext{log}_g 7 + ext{log}_g 4 + rac{1}{2}$, find $y$ in terms of $a$ - AQA - A-Level Maths: Pure - Question 7 - 2018 - Paper 3
Step 1
Find $y$ in terms of $a$
Answer
We start with the equation:
ext{log}_g y = 2 ext{log}_g 7 + ext{log}_g 4 + rac{1}{2}
Using properties of logarithms, we can combine the log terms:
ext{log}_g y = ext{log}_g(7^2) + ext{log}_g(4) + rac{1}{2}
Now, applying the product property of logarithms:
ext{log}_g y = ext{log}_g(49) + ext{log}_g(4) + rac{1}{2} ext{log}_g y = ext{log}_g(196) + rac{1}{2}
We can convert rac{1}{2} into logarithmic form:
rac{1}{2} = ext{log}_g( ext{sqrt}(g))
Thus, we can represent this as:
Hence,
Step 2
Explain what is wrong with the student's solution
Answer
The student's solution incorrectly concludes that x = rac{3}{2} is valid. However, the value rac{3}{2} should be rejected because it is not possible to evaluate ext{log}_g rac{3}{2}. Since this value is derived from determining the logarithm of a fraction, this specific case leads to the conclusion that the logarithm is not defined for this input.