Mathematics

Mathematics

Problem

Rainwater collected in a laboratory sample has a hydrogen–ion concentration that may vary between 10−6 mol L−110^{-6}\text{ mol L}^{-1} and 10−3 mol L−110^{-3}\text{ mol L}^{-1}. Chemists use the quantity p(x)=−log⁡10(x),p(x)= -\log_{10}(x), where xx is the hydrogen–ion concentration in moles per litre, to indicate the acidity of the water. Answer the following with reference to the above situation. a. (i) Identify the input variable and the output variable in this context. State with reason whether pp associates exactly one output value with every admissible input value. b. (ii) Determine the domain and the range of the function pp for the given interval of concentrations. c. (iii) Is it possible, under the stated physical conditions, for the value of p(x)p(x) to be negative? Justify your answer using the definition of the range obtained in part (ii).
Intuitive explanation

The pH scale compresses a huge range of hydrogen-ion concentrations into a manageable 0–14 range using logarithms. Since concentration varies exponentially (from 10−610^{-6} to 10−310^{-3}), the logarithm function p(x)=−log⁡10(x)p(x) = -\log_{10}(x) flips and linearizes that exponential spread—so you'll find the pH varies between just two simple integers. Understanding how logarithms invert powers of 10 is the key to moving fluently between concentration and acidity.

Intuitive explanation

The pH scale compresses a huge range of hydrogen-ion concentrations into a manageable 0–14 range using logarithms. Since concentration varies exponentially (from 10−610^{-6} to 10−310^{-3}), the logarithm function p(x)=−log⁡10(x)p(x) = -\log_{10}(x) flips and linearizes that exponential spread—so you'll find the pH varies between just two simple integers. Understanding how logarithms invert powers of 10 is the key to moving fluently between concentration and acidity.

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