# Engaging students: Computing logarithms with base 10

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Jonathan Chen. His topic, from Precalculus: computing logarithms with base 10.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Computing logarithms with base 10 can appear in many scientific applications for word problems. To define the acidity or alkalinity of a substance, Chemists use the formula $pH = \log [H^+]$. “[H+] is the hydrogen ion concentration that is measured in moles per liter” (Stapel, n.d.). We know lemon juice is acidic because the pH value is less than 7. We know bleach is basic because the pH value is greater than 7. When a pH value is equal to 7, the solution is neutral. An example of something neutral would be pure water. Teacher can create word problems based on the information given about a liquid solution. Noise can be measured in decibels. The formula used to measure the strength of a sound is $dB = 10 \log(I \div I_0)$. “I0 is the intensity of ‘threshold sound,’ or sound that can be barely be perceived” (Stapel, n.d.). Teachers can create word problems based on the defined terms of how many times more intense a sound is than the threshold sound. Similar problems with the topic of computing logarithms can be made involving earthquake intensity.

How can this topic be used in your students’ future courses in mathematics or science?

As shown in the above answer, this topic can reappear in student’s future science course in the topic of pH levels, earthquake intensity, or “loudness” measured in decibels. In order to find the pH levels, [H+] concentration, or the [OH] concentration you may need to know how to calculate logarithms with base 10 when dealing with the equation $pH = \log [H^+]$. Similar things can be said about measuring “loudness” and earthquake intensity. Their formulas involve calculating logarithms with base 10. Other future topics students may encounter in mathematics are logarithmic functions, Euler’s number, natural log, and logarithm rules. While not all of these future topics are strongly related to the topic of calculating logarithms with base 10, they can be loosely connected to where the practice of calculating logarithms with base 10 makes it easier to understand and do things related to the future topics. With the topic of logarithmic rules, it can help better simply and calculate with logarithms with base 10.

Calculating logarithms with base 10 has been around since 1614. John Napier invented logarithms and ever since then small additions have been made. Additions such as a logarithmic table made it easier to solve logarithmic problems. The logarithmic tables are similar to the multiplication tables elementary schoolers memorize to calculate simple multiplication faster for their future problems. Many mathematicians made their contributions to add more to the logarithmic table to the point where the calculations reached up to 200,000. Aside from the logarithmic tables, there were other methods to calculate logarithms with base 10 such as the slide rule. It was also possible to memorize the values of the logs with base 10 of 1 through 10 and use the logarithmic rules to calculate bigger values. Because

$\log 400 = \log(100 \times 4) = \log 4 + \log 100$

by expansion and logarithmic rules, people can solve this problem my memorizing that $\log 4 = 0.602$ and knowing that $\log 100 = \log 10^2 = 2$. Knowing this makes the equation more clear to recognize and easier to solve by hand. Calculating logarithms with base 10 were used extensively until the creation of the calculator made it easier to calculate anything, including logarithms.

References

“The Log Log Duplex Trig” “Slide Rule”. (n.d.). Retrieved from Web Archive: https://web.archive.org/web/20090214020502/http://www.mccoys-kecatalogs.com/K%26EManuals/4081-3_1943/4081-3_1943.htm

Bourne, M. (n.d.). 4. Logarithms to Base 10. Retrieved from Interactive Mathematics: https://www.intmath.com/exponential-logarithmic-functions/4-logs-base-10.php