Real-World Applications of Logarithmic Functions

Before beginning this tutorial, you may wish to review the tutorial on Arithmetic With Numbers In Scientific Notation.

Example 1

In chemistry, a solution’s pH is defined by the logarithmic equation , where t is the hydronium ion concentration in moles per liter. We usually round pH values to the nearest tenth.

a. Find the pH of a solution with hydronium ion concentration 4.5 x 10^-5

b. Find the hydronium ion concentration of pure water, which has a pH of 7.

Solutions

If t=4.5 x 10^-5, then p(t)= -log₁₀(4.5 x 10^-5)= -(log₁₀4.5 + log₁₀10^-5)= -(log₁₀4.5 + (-5)(log₁₀10))= -(.6532+-5)= -(-4.3468)=4.3.

Since water has a pH of 7, we know 7=-log₁₀t and so 7=log₁₀t^-1; thus 10⁷= t^-1, and so the hydronium ion concentration of water is t=10^-7 moles per liter.

Example 2

Measuring decibels of sound

The loudness of sound is measured in units called decibels. These units are measured by first assigning an intensity I0 to a very soft sound (which is called the threshold sound). The sound we wish to measure is assigned an intensity I, and we measure the decibel rating d of this sound with the equation .

Find the decibel rating of a sound with intensity 5000I₀.

If a sound has a decibel rating of 85, how much more intense is it than the threshold sound?

Solutions

            a. = decibels
            b. , where the sound in question is k times as intense as
                  the threshold sound. Thus , and so the sound is
                  times as intense as the threshold sound.