Question

This is the first question. I have completed the graph which is attached below. Create a graph of the pH function either by hand or using technology. Locate on your graph where the pH value is 0 and where it is 1. This is the second question The pool maintenance man forgot to bring his logarithmic charts, and he needs to raise the amount of hydronium ions, t, in the pool to 0.50. To do this, he can use the graph you created. Use your graph to find the pH level if the amount of hydronium ions is raised to 0.50. Then, convert the logarithmic function into an exponential function using y for the pH.

Answer 1

The Solution:

From the given graph, we have that

`\begin{gathered} \text{ When t=0, p=}2 \\ \text{ When t=0.1, p=1} \end{gathered}`

We need to find the logarithmic function that describes the situation.

`\begin{gathered} p=\log _at \\ \text{ Where} \\ p=PH\text{ function} \\ t=\text{amount of hydronium ions} \end{gathered}`

So, the logarithmic function is

`p=\log _at`

To find the value of a, we shall use the initial values:

`\begin{gathered} p=1\text{ when t=0.1} \\ \text{Substituting, we get} \\ 1=\log _a0.1 \\ a^1=0.1 \\ a=0.1 \end{gathered}`

Thus, the logarithmic function is

`p=\log _(0.1)t`

Writing the logarithmic function as an exponential function, we get

`t=0.1^p`

The pool maintenance man wants to raise the amount of hydronium ions to 0.50 , that is, t=0.50, find the PH level, that is, the value of p.

`\begin{gathered} \text{ When t=0.50},\text{ find p} \\ \text{Substituting into the function, we get} \\ 0.50=0.1^p \\ \text{ Taking the logarithm of both sides, we get} \\ \log _{}0.50=\log _{}0.1^p \\ \log _{}0.50=p\log _{}0.1 \end{gathered}`
`p=\frac{\log _{}0.50}{\log _{}0.1}=0.3010`

Therefore, the PH level will be 0.3010.