Logs and Natural Ln

 I’m in the chapter dealing with exponential functions in Algebra II, specifically ln and log.  I have one student who is always questioning its use in real life.  My first example is always the Richter scale but I need more than that.  First of all, logarithmic functions are used to help solve exponential equations such as the Richter scale, calculating how bright a star is, finding pH or the decibel level of sound.

The Richter scale is calculated as the log (the amplitude of the wave/amplitude of the smallest detectable wave)  so if there is an earthquake and it is found to be 392 times greater than the smallest detectable wave, the equation becomes log (392/1) or log 392 = 2.59…. Or 2.6 if rounded.  Although 392 times larger sounds like a lot, it is really a small earthquake.  

The equation to determine the decibel levels for sound is almost the same basic equation.  It is the decibel rating = 10log(power or intensity of sound/the weakest sound heard by the human ear). So if the students in the  classroom are quiet you have a reading of 10^-7 and if the room is empty, the reading is 10^-12 so the equation would be Decibel rating = 10 Log (10^-7/10^-12) which becomes 10Log10^5 or 10*5 = 50 decibels.

As for finding pH, the equation is -log[H^+] or -Log[ the concentration of hydrogen. If you have a concentration of .0025 for the HCL solution.  Then you simply use the .0025 in the equation so it looks like pH = -log[0.0025] and the pH is 2.6.

In addition, ln and log are used to find missing values in both exponential growth and decay, interest, populations and so much more. Unfortunately, most of my students do not think they will every use this even after I shared with them the time I sat down to calculate whether financing via the credit union or the dealership was better. They figure I did that because I am into math.

For all the time, I’ve taught math, I’ve never had a book that took time to look at the different applications of logs other than in a purely theoretical way. I think that the next time I teach logs, I’m going to take time to have students explore the equations used for earthquakes, sound levels, etc and create a small sticky note in the Padlet application.

Students could research the topic so they could explain the basic equation and then provide examples for using the equation to find different things.  For instance, have students find the rating of the earthquake just like I showed in my example and then give them the rating of an earthquake such as 7.2 and have them determine the intensity of the wave.

This type of activity would put  their learning into context and provide real situations where the math is really used rather than being so isolated.  Let me know what you think, I’d love to hear.  Have a great day.