Let’s start by solving the easy (or maybe not so much) logarithmic problems from last week:

2 = √2 x √2 log 2 = 2 log √2 = 0,3 log √2 = 0,3/2 = 1,5

log 1/2 = log 1 – log 2 = 0 – 0,3 = –0,3

log 80 = log (8 x 10) = log 2³ + log 10 = 3 x 0,3 + 1 = 1,9

log a = 1,3 a = 20

log b = 2,15 b = 100√2

log c = 0,7 c = 5

### What is dough?

Since in the previous weeks we have talked about the meter and the second, the units of length and time, from which some others are derived (of surface, volume, speed, etc.), it is inevitable to talk about the third basic unit, which is the kilogram, the unit of mass according to the International System of Units (SI), currently in force in almost all the countries of the world.

We all have an intuitive idea of mass as the amount of matter in an object; but this notion is misleading because it is imprecise, since it can be confused with the amount of substance. And, on the other hand, it is also common to confuse mass with weight, since we live immersed in the Earth’s gravitational field and we perceive the mass of objects (and our own body) above all by their weight. At a colloquial level, it is common to confuse mass and weight, as when it is said that something weighs 10 kilograms, when in reality it should be said 10 kiloponds: the kilogram is the unit of mass, and the kilopond is the force with which the Earth attracts an object of 10 kilograms of mass.

Mass is defined in relation to inertia: it is that property of a physical object that determines its acceleration when subjected to a force (according to the formula F = ma).

Regarding the quantity of substance, another of the fundamental magnitudes is the number of elementary entities (usually atoms or molecules, although it can also be ions, electrons or other particles). The unit is the mole, which is the amount of substance that has a number of elemental entities equal to the number of atoms in 12 grams of carbon-12. That number is known as Avogadro’s number, approximately 6.022 x 10²³.

A mole of an element is its atomic weight expressed in grams and, therefore, in a mole of any element there is the same number of atoms (Avogadro’s number). That is why moles are used above all to adjust chemical reactions, since they indicate the quantities of the different elements that take part in a reaction or a formula; thus, since carbon dioxide is CO₂, one mole of CO₂ (28 grams) is formed by combining one mole of carbon (12 grams) with two moles of oxygen (2 x 8 = 16 grams). This is why the amount of substance is also called the “chemical amount”.

And, speaking of masses and weights, a couple of heavy problems (or one in two parts):

1. A grocer is made from a thick chain of thirteen links, each of which weighs one kilogram. He is very happy, as he now has thirteen one kilo weights with which he can weigh on his scale, in a single weight, any quantity between 1 and 13 kilos. To do this, obviously, you have to open some of the links to divide the chain into the appropriate pieces. How many links does he have to open at least? (To raise the grade, generalize the problem for a chain of n links).

2. A miner asks the shopkeeper to prepare 5 kilos of gunpowder for him by mixing the corresponding amounts of potassium nitrate, sulfur, and powdered coal. The shopkeeper, who has no more weights than the links in the chain, tells him that, at least, he will have to buy 8 kilos of gunpowder. Knowing that black powder contains the same amount of sulfur as carbon and more nitrate than sulfur, what is its composition?

**Carlo Frabetti ***is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.*

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Eddie is an Australian news reporter with over 9 years in the industry and has published on Forbes and tech crunch.