Over the past several weeks, some of the seniors at Roberto Clemente Community Academy have been studying a unit on exponential functions. The classes have been examining these functions through different lenses, including real world modelings, such as modeling population growth (and decline) and interest rates. A key component of working with exponential functions are their inverses: logarithms.
This week, the Clemente College Algebra students engaged in an investigation into logarithmic rules. The seniors were tasked with finding general rules whereby two or more logarithms could be combined. The Wildcats had to draw on prior knowledge to aid in discovering the logarithmic rules.
The class started its investigation by gathering data in the form of various logarithmic values. Students looked at the tables they created and had to coax out general trends and patterns. While certain patterns were easy to detect, some students had difficulty in identifying other trends. Nonetheless, the students persisted, working with one another to create proper conjectures that led to useful insights.
After the data analysis, the students engaged in some problem solving where they were tasked with finding the value of the sums of logarithms. In order to do this, the students used logarithm rules to find solutions. Once, completed, the class had to review their work and devise a rule that generalized their findings. The seniors were able to formulate the product rule for logs through their analyses. Moreover, some of the students came to these conclusions after solving only one problem!
Next, the students continued their examination of logs by investigating what happens when logs are subtracted. Again, the students used the logarithm rules to find the answers to log subtraction and were able to state the quotient rule of logs. At this point, some of the class had made the connection between the log rules and the exponent rules they had worked out earlier in the year. Using this realization and some more problem solving, the Wildcats quickly formulated the power rule of logs. Overall, the students did an exceptional job in dealing with a new, unfamiliar topic and persevered by trusting their own abilities, calling on prior knowledge, and by using problem-solving skills.