6.4: Logarithmic Functions and Applications

1. For these questions, you need to know what the graph of \(y=e^x\) and \(y=\ln x\) look like. You should know these graphs, but if you need a reminder, use a calculator or online graphing tool.

[(a)] What is the equation of the asymptote of the graph of \(y=e^x\) ?

 

 

[(b)] What is the equation of the asymptote of the graph of \(y=\ln x\) ?

 

 

[(c)] Explain in a sentence the connection between these two asymptotes.

 

 

2. The pH of a solution is a measure of its acidity or basicity. It is computed from the solution's hydrogen ion concentration \(\left[H^{+}\right]\), in moles per liter, by

\[\mathrm{pH}=-\log \left[H^{+}\right] .\]

[(a)] Use a calculator to find, the hydrogen ion concentration of each of the following. How much precision do you need for reasonable answers?

[i.] ... hydrochloric acid, with a pH of 0.

 

 

[ii.] ...lemon juice, with a pH of 2.

 

 

[iii.] ... urine, with a pH of 6.5.

 

 

[iv.] ... household bleach, with a pH of 13.5.

[(b)] Considering your answers to the previous questions, why is the logarithm used in the definition of pH? Explain in a sentence or two.
 

 

 

3. Each of the following graphs is an exponential or a logarithmic function. Give an expression for each function.

 

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