**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__

**pH**

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*, then^{-5}*p(t)= -log*._{10}(4.5 x 10^{-5})= -(log_{10}4.5 + log_{10}10^{-5})= -(log_{10}4.5 + (-5)(log_{10}10))= -(.6532+-5)= -(-4.3468)=4.3

- Since water has a pH of
*7*, we know*7=-log*and so_{10}t*7=log*; thus_{10}t^{-1}*10*, and so the hydronium ion concentration of water is^{7}= t^{-1}*t=10*moles per liter.^{-7}

__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
*5000I*._{0}

- 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.