Applications of Common Logarithms

Examples Involving pH

In chemistry, the pH of a solution is defined as  LaTeX: pH=-\log\left[H_3O^+\right],\:pH=−log[H3O+],where LaTeX: \left[H_3O^+\right]\:[H3O+]is the hydronuim ion concentration in moles per liter. The pH value is a measure of the acidity or alkalinity of a solution. Pure water has pH 7.0, substances with pH values greater than 7.0 are alkaline, and substances with pH values less than 7.0 are acidic. It is customary to round pH values to the nearest tenth.

  1. Find the pH of a solution with LaTeX: \left[H_3O^+\right]=2.5\times10^{-4}.[H3O+]=2.5×10−4.
  2. Find the hydronium ion concentration of a solution with pH = 7.1.

  3. Wetlands are classified as bogs, fens, marshes, and swamps based on pH values. A pH value between 6.0 and 7.5 indicates that the wetland is a “rich fen.” When the pH is between 3.0 and 6.0, it is a “poor fen,” and if the pH falls to 3.0 or less, the wetland is a “bog.”

    Suppose that the hydronium ion concentration of a sample of water from a wetland is LaTeX: 6.3\times 10^{-5} 6.3×10−5. How would this wetland be classified?

Solutions

  1. Find the pH of a solution with LaTeX: \left[H_3O^+\right]=2.5\times10^{-4}.[H3O+]=2.5×10−4.
    LaTeX: pH=-\log\left[H_30^+\right]=-\log\left(2.5\times10^{-4}\right)=-\left(\log2.5+\log10^{-4}\right)pH=−log[H30+]=−log(2.5×10−4)=−(log2.5+log10−4)
    LaTeX: =-\left(0.3979-4\right)=-0.3979+4\approx3.6=−(0.3979−4)=−0.3979+4≈3.6
    Note: We used the equality symbol, =, rather than the approximate equality symbol, ≈, when replacing log 2.5 with 0.3979. This is often done for convenience, despite the fact that most logarithms used in applications are indeed approximations.
  2. Find the hydronium ion concentration of a solution with pH = 7.1.
    LaTeX: pH=-\log\left[H_30^+\right]pH=−log[H30+]
    LaTeX: 7.1=-\log\left[H_30^+\right]7.1=−log[H30+]
    LaTeX: -7.1=\log\left[H_30^+\right]−7.1=log[H30+]
    LaTeX: \left[H_30^+\right]=10^{-7.1}[H30+]=10−7.1
    LaTeX: \left[H_30^+\right]\approx7.9\times10^{-8}[H30+]≈7.9×10−8

  3. Suppose that the hydronium ion concentration of a sample of water from a wetland is LaTeX: 6.3\times 10^{-5} 6.3×10−5. How would this wetland be classified?
    LaTeX: pH=-\log\left[H_3O^+\right]=-\log\left(6.3\times10^{-5}\right)pH=−log[H3O+]=−log(6.3×10−5)
    LaTeX: =-\left(\log6.3+\log10^{-5}\right)=-\log6.3-\left(-5\right)=-\log6.3+5\approx4.2=−(log6.3+log10−5)=−log6.3−(−5)=−log6.3+5≈4.2

Example Involving the Loudness of Sound

The loudness of sounds is measured in decibels. We first assign an intensity of LaTeX: I_0I0 to a very faint threshold sound. If a particular sound has intensity LaTeX: II, then the decibel rating LaTeX: dd of this louder sound is given by the following formula.

LaTeX: d=10\log\frac{I}{I_0}.d=10logII0.

Find the rating in decibels of a sound with intensity 10,000LaTeX: I_0I0.

Solution

LaTeX: d=10\log\frac{10,000I_0}{I_0}=10\log10,000=10\left(4\right)=40d=10log10,000I0I0=10log10,000=10(4)=40

The sound has a decibel rating of 40.