# Square roots and logarithms without a calculator (Part 9)

This post is not really about finding square roots but continues Part 8 of this series. Continuing the theme of this series, let’s go back in time to when scientific calculators were not invented… say, 1850.

This is a favorite activity that I use when teaching logarithms to precalculus students. I begin by writing the following on the board, in three or four columns:

$\log_{10} 1$

$\log_{10} 2 \approx 0.301$

$\log_{10} 3 \approx 0.477$

$\log_{10} 4$

$\log_{10} 5$

$\log_{10} 6$

$\log_{10} 7$

$\log_{10} 8$

$\log_{10} 9$

$\log_{10} 10$

$\log_{10} 11$

$\log_{10} 12$

$\log_{10} 13$

$\log_{10} 14$

$\log_{10} 15$

$\log_{10} 16$

$\log_{10} 17$

$\log_{10} 18$

$\log_{10} 19$

$\log_{10} 20$

$\log_{10} 30$

$\log_{10} 40$

$\log_{10} 50$

$\log_{10} 60$

$\log_{10} 70$

$\log_{10} 80$

$\log_{10} 90$

$\log_{10} 100$

In other words, I tell the answer to only $\log_{10} 2$ and $\log_{10} 3$. The challenge: fill in the rest without a calculator.

In my classes, we found these logarithms by large-group discussion. However, there’s no reason why this couldn’t be done by dividing a class into small groups and letting the groups collaborate. Indeed, I suggested this idea to a former student who was struggling to come up with an engaging activity involving logarithms for an Algebra II class that she was about to teach. She took this idea and ran with it, and she told me it was a big hit with her students.

I provide a thought bubble if you’d like to think about it before I give the answers.

Step 1. Three of these values — $1$, $10$, and $100$ — can be found exactly since they’re powers of $10$.

Step 2. Most of the others can be found by using the laws of logarithms for products, quotients, and powers involving $2$, $3$, and $10$. For example,

$\log_{10} 9 = \log_{10} 3^2 = 2 \log_{10} 3 \approx 2 \times 0.477 = 0.954$

$\log_{10} 20 = \log_{10} 2 + \log_{10} 10 = 1.301$

$\log_{10} 5 = \log_{10} 10 - \log_{10} 2 = 0.699$.

Of this group, usually $\log_{10} 5$ is the hardest for students to recognize.

Step 3 (optional). A few of the logarithms, like $\log_{10} 7$, cannot be determined in terms of $\log_{10} 2$ and $\log_{10} 3$. But they can be approximated to reasonable accuracy with a little creativity. For example,

$\log_{10} 7 = \log_{10} \sqrt{49} = \frac{1}{2} \log_{10} 49 \approx \frac{1}{2} \log_{10} 50 = \frac{1}{2} (1.699) = 0.850$.

For a really good approximation, we use the fact that $7^4 = 2401 \approx 2400$.

$\log_{10} 7 = \frac{1}{4} \log_{10} 2401 \approx \frac{1}{4} \log_{10} 2400 = \frac{1}{4} (3 \log_{10} 2 + \log_{10} 3 + \log_{10} 100) = 0.845$.

To approximate $\log_{10} 17$, we could use the fact that $(16-1) \times (16 + 1) = 16^2-1$, or $15 \times 17 = 255 \approx 2^8$. So

$\log_{10} 17 \approx 8 \log_{10} 2 - \log_{10} 15 = 8 \log_{10} 2 - \log_{10} 3 - \log_{10} 5 = 1.232$

Naturally, any and all of the above results can be confirmed with a scientific calculator.

In my opinion, here are some of the pedagogical benefits of the above activity.

1. This activity solidifies students’ knowledge about the laws of logarithms. The laws of logarithms become less abstract, changing from $\log_{10} xy = \log_{10} x + \log_{10} y$ into something more tangible and comfortable, like positive integers.

2. Hopefully the activity will demystify for students the curious decimal expansions when a calculator returns logarithms. In other words, hopefully the above activity will help

3. The activity should promote some understanding of the values of base-10 logarithms. For example, $0 \le \log_{10} x < 1$ for $1 \le x < 10$ and $1 \le \log_{10} x < 2$  for $10 \le x < 100$.

4. Students should see that, for large $x$, $\log_{10}(x+1)$ is not much larger than $\log_{10} x$. This is another way of saying that the graph of $y = \log_{10} x$ increases very slowly as $x$ increases. So this should provide some future intuition for the graphs of logarithmic functions.

5. The values of $\log_{10} 2, \log_{10} 3, \dots, \log_{10} 9$ are used to construct the unevenly-spaced lines and/or tick marks in log-log graphs and log-linear graphs (which are standard plotting options on many scientific calculators).