## Logarithms and a summer break when the calculator did not work

November 15, 2019 by Chetana Badala Viswanatha

When Nandu was in grade seven, her pastime during commute was to observe high school students. The only other way, apart from the apparent difference in uniform, to spot a high school senior was the peeking of Clark’s Tables from their bags. The maroon book was intimidating with the statement “Science Data Book” printed on the cover.

She finally acquired a copy of it after relentlessly badgering her father.
The book lived up to its title and comprised of mostly incomprehensible tables. Skimming through the book, Nandu found two terminologies intriguing, trigonometry and logarithms. Trigonometry seemed slightly more familiar due to association with triangles.

Nandu had to wait until the summer break to get a hint about logarithmic tables. The calculator stopped working, and she struggled with the arithmetic of large numbers. Nandu’s mother recalled using logarithmic tables to compute the products of large numbers. Unable to contain her curiosity any longer, Nandu tried to get a grasp on the concept of logarithms. Building on her mother’s hint that she would need “common logarithms” for arithmetic, Nandu only focused on common logarithms.

It was straightforward to see that $10\times10\times10$ is the same as $10^3$ or $1000$ or $10\times10$ or $10\times100$. Similarly, it was easy to follow the reasoning –

$10\times 10\times 10 \times \cdots \times \cdots n = 10^n$

$10\times10\times10 \cdots \times \cdots m \times 10\times10\times10 \cdots \times \cdots n = 10^m\times 10^n=10^{m+n}$

Now it was time to bring in the logarithmic notation.

If $10^m=x$, then $\log_{10}x=m$

The advantage of using logarithms for arithmetic comes from exploiting the properties of the exponential, where multiplication gets reduced to addition.

Nandu found the reasoning comprehensible. However, there was still a significant missing link. How could she use this knowledge for arithmetic calculations?

Like a sudden bolt of lightning, it struck her! The missing jigsaw piece had been there all along. The journey in school had taught her all the necessary concepts. But she had never looked at it in all its entirety.
It dawned on her that –

$2\times6=12.0$

$20\times60=1200.0$

$0.2\times0.06=0.012$

The numbers in the product only differ in the position of the decimal point! The reason becomes more evident if the numbers mentioned above are written in the following manner.

$2 \times 10^0 \times6 \times 10^0=1.2 \times 10^1$

$2 \times 10^1 \times6 \times 10^1=1.2 \times 10^3$

$2 \times 10^{-1}\times6 \times 10^{-2}=1.2 \times 10^{-2}$

It would then make complete sense to compute such products and tabulate them. The table serves as an elementary calculator for positive numbers. This is the general idea behind common logarithmic tables.

With a satisfying feeling of being in the high school club, Nandu thoroughly enjoyed her summer break.

If you are further interested in logarithms like Nandu, I strongly recommend this Vihart video.

Post written by Chetana Badala Viswanatha

Chetana Badala Viswanatha (Chetana B V) is a freelance science communicator with a passion for science writing. She holds a Master's degree with a specialization in Quantum Chemistry from RUB, Germany. She is currently pursuing a PhD in Experimental Physics at TUK, Germany. Her master thesis work on the theoretical calculation of circularly polarized phosphorescence is published (DOI: 10.1039/C8CP03385D) in the 2018 PCCP HOT Articles. She believes in crisp and simple explanation of scientific concepts.