Logarithmic Word Problems
Purplemath
What are logarithm word problems?
Logarithmic word problems, in my experience, generally involve either evaluating a given logarithmic equation at a given point, or else solving an equation for a given variable; they're pretty straightforward.
What real-world problems use logarithms?
The classic real-world contexts for logarithm word problems are the measurement of acidity or alkalinity (that is, the measurement of pH), the measurement of sound (in decibels, or dB), and the measurement of earthquake intensity (on the Richter scale), among other uses (link).
Note: While log-based word problems are, in my experience, pretty straightforward, their statements tend to be fairly lengthy. Expect to have to plow through an unusual amount of text before they get to the point.
- 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+]. In other words, this exercise, despite all the verbiage, is just plug-n-chug.
Since no base is specified, I will assume that the base for this logarithm is 10, so that this is the so-called "common" log. (I happen to know that 10 is indeed the correct base, but they should have specified.)
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...
This value is less than 7, so the apple juice 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...
This value is more than 7, so the ammonia is basic.
(a) The juice is acidic with a pH of about 3.5, and
(b) the ammonia is basic with a pH of about 8.9.
When a logarithm is given without a base being specified, different people in different contexts will assume different bases; either 10, 2, or e. Ask now whether or not bases will be specified for all exercises, or if you're going to be expected to "just know" the bases for certain formulas, or if you're supposed to "just assume" that all logs without a specified base have a base of... [find out which one].
- "Loudness" is measured in decibels (abbreviated as dB). The formula for the loudness of a sound is given by dB = 10×log[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:
...about 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 practicing at the rifle range?
I need to evaluate the decibel equation at I = (2.5 × 1013)I0:
In other words, the squirrel gun 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.
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 of the mystery quake into a Richter rating by evaluating the Richter function at I = 989I0:
A Richter rating of about 3 is not high enough to have been a moderate quake.
The event was probably just a big truck.
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