Logarithmic Functions: Apply It

  • Convert between logarithmic to exponential form.
  • Evaluate logarithms.
  • Use common and natural logarithms

Water Quality Assessment for Safe Swimming

Maria is an environmental scientist working for the county health department. She tests water quality at public beaches and swimming areas to ensure they’re safe for recreation. One of her key measurements is pH level, which indicates how acidic or basic the water is.

The pH scale ranges from 0 (very acidic) to 14 (very basic), with 7 being neutral. Safe swimming water should have a pH between 7.2 and 7.8.

Maria needs to convert between pH readings and actual hydrogen ion concentrations to properly assess water safety and communicate results to the public.

The relationship between pH and hydrogen ion concentration follows a logarithmic pattern: [latex]\text{pH} = -\log[H^+][/latex]. Converting between logarithmic and exponential forms allows us to move between the convenient pH scale and the actual chemical concentrations that determine water safety.

To convert from logarithmic form to exponential form:

  1. Identify the logarithmic equation: [latex]\text{pH} = -\log[H^+][/latex]
  2. Isolate the logarithm: [latex]-\text{pH} = \log[H^+][/latex]
  3. Convert to exponential form: [latex][H^+] = 10^{-\text{pH}}[/latex]

Maria tests water at Sunset Beach and finds a pH of 8.2. What is the hydrogen ion concentration?

Maria receives lab results showing a water sample has a hydrogen ion concentration of [latex][H^+] = 1.0 \times 10^{-6}[/latex] moles per liter. What is the pH?

When evaluating [latex]\log(10^n) = n[/latex], remember that the logarithm asks “to what power must 10 be raised to get this number?”