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Given that $a > 0$, determine which of these expressions is not equivalent to the others - AQA - A-Level Maths Mechanics - Question 1 - 2019 - Paper 1

Question 1

Given that $a > 0$, determine which of these expressions is not equivalent to the others. Circle your answer. - $-2\log_{10}\left(\frac{1}{a}\right)$ - $2\log_{10... show full transcript

Worked Solution & Example Answer:Given that $a > 0$, determine which of these expressions is not equivalent to the others - AQA - A-Level Maths Mechanics - Question 1 - 2019 - Paper 1

Step 1

$-2\log_{10}\left(\frac{1}{a}\right)$

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Answer

Using the logarithmic property, this expression can be simplified:

$-2\log_{10}\left(\frac{1}{a}\right) = -2(-\log_{10}(a)) = 2\log_{10}(a)$

Thus, it is equivalent to $2\log_{10}(a)$ .

Step 2

$2\log_{10}(a)$

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Answer

This expression is already in a simplified form and represents itself.

Step 3

$\log_{10}(a^{2})$

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Answer

According to logarithmic properties:

$\log_{10}(a^{2}) = 2\log_{10}(a)$

This is also equivalent to $2\log_{10}(a)$ .

Step 4

$-4\log_{10}(\sqrt{a})$

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Answer

Using the logarithmic property for square roots:

$-4\log_{10}(\sqrt{a}) = -4\left(\frac{1}{2}\log_{10}(a)\right) = -2\log_{10}(a)$

This expression is different as it simplifies to $-2\log_{10}(a)$ , which does not match the others.

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