Logarithmic Word Problems (page 1 of 3)

Sections: Log-based word problems, exponential-based word problems

Logarithmic word problems, in my experience, generally involve evaluating a given logarithmic equation at a given point, and solving for a given variable; they're pretty straightforward. On the other hand, exponential word problems tend to be much more involved, requiring, among other things, that the student first generate the exponential equation, and perhaps then also find the value of one of the variables before beginning to answer the actual question. Since log problems are typically simpler, I'll start with them.

• Chemists define the acidity or alkalinity of a substance according to the formula
  "pH = –log[H⁺]" where [H⁺] is the hydrogen ion concentration, measured in moles per liter. Solutions with a pH value of less than 7 are acidic; solutions with a pH value of greater than 7 are basic; solutions with a pH of 7 (such as pure water) are neutral.

a) Suppose that you test apple juice and find that the hydrogen ion concentration is [H⁺] = 0.0003. Find the pH value and determine whether the juice is basic or acidic.

b) You test some ammonia and determine the hydrogen ion concentration to be [H⁺] = 1.3 × 10^–9. Find the pH value and determine whether the ammonia is basic or acidic.

In each case, I need to evaluate the pH function at the given value of [H⁺].

a) In the case of the apple juice, the hydrogen ion concentration is [H⁺] = 0.0003, so:

pH = –log[H⁺] = –log[0.0003] = 3.52287874528...

...which is less than 7, so this is acidic.

b) In the case of the ammonia, the hydrogen ion concentration is [H⁺] = 1.3 × 10^–9, so:

pH = –log[H⁺] = –log[1.3 × 10^–9] = 8.88605664769...

...which is more than 7, so this is basic.

The juice is acidic with a pH of about 3.5,
and the ammonia is basic with a pH of about 8.9.

• "Loudness" is measured in decibels. The formula for the loudness of a sound is given by "dB = 10log[ I ÷ I₀ ]" where I₀ is the intensity of "threshold sound", or sound that can barely be perceived. Other sounds are defined in terms of how many times more intense they are than threshold sound. For instance, a cat's purr is about 316 times as intense as threshold sound, for a decibel rating of:

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Db = 10log[ I ÷ I₀ ]
    = 10log[ (316 I₀) ÷ I₀ ]
    = 10log[ 316 ]
    = 24.9968708262...,

...or  25 decibels.

Considering that prolonged exposure to sounds above 85 decibels can cause hearing damage or loss, and considering that a gunshot from a .22 rimfire rifle has an intensity of about I = (2.5 ×10¹³)I₀, should you follow the rules and wear ear protection when relaxing at the rifle range?


I need to evaluate the decibel equation at I = (2.5 ×10¹³)I₀:

Db = 10log[ I ÷ I₀ ]
    = 10log[ (2.5 ×10¹³)I₀ ÷ I₀ ]
    = 10log[2.5 ×10¹³]
    = 133.979400087...

In other words, my rifle creates a noise level of about 134 decibels. Since this is well above the level at which I can suffer hearing damage,

I should follow the rules and wear ear protection.

• Earthquake intensity is measured by the Richter scale. The formula for the Richter rating of a given quake is given by "R = log[ I ÷ I₀ ]" where I₀ is the "threshold quake", or movement that can barely be detected, and the intensity I is given in terms of multiples of that threshold intensity.   Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved

You have a seismograph set up at home, and see that there was an event while you were out that had an intensity of I = 989I₀. Given that a heavy truck rumbling by can cause a microquake with a Richter rating of 3 or 3.5, and "moderate" quakes have a Richter rating of 4 or more, what was likely the event that occurred while you were out?

To determine the probable event, I need to convert the intensity to a Richter rating by evaluating the Richter function at I = 989I₀:

R = log[ I ÷ I₀ ]
    = log[ 989I0 ÷ I₀ ]
    = log[989]
    = 2.9951962916...

A Richter rating of about 3 is not a high enough rating to have been a moderate quake;

the event was probably just a big truck going too fast over the speed humps in my neighborhood.

A closely-related class of exercises involves exponential equations....

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