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# House of Maths School Workshops Primary & Secondary in Dorset & South - DOGARITHMS

### WHAT’S THE POINT OF LOGARITHMS?

On meeting this adorable litter of ten Dalmation Puppies the other day, I quickly spotted (ahem) the need for Dogarithms – the canine equivalent of logarithms. The conversation with dog breeder Maxine went something like this:

HOM (House Of Maths): Oh they are adorable – please may I hold one??

M (MAXINE): Yes of course, here’s Copper.

HOM: He’s so big already!! It seems impossible that they could all have fitted inside Mum Inca just a few weeks ago.

M: Yes they have grown so much!

HOM: How often do they double in size? [This seemed like a natural question for one to ask, whether a mathematician or not!].

M: Well Copper is now 3kg, but his birthweight five weeks ago was just 265 grams.

A few moments later, Copper was transferred to my daughter’s arms (see totally smitten photo!) and I was brandishing my phone, calculator app at the ready!

Now then: 3kg=3000g; dividing this by 265g, we get that Copper has grown 11.32 times as big since birth. So how many times has he doubled in size? More than three, because $2^3=8<11.32$, but not as many as four times, because $2^4=16>11.32$. So how many times do we multiply two by itself to get 11.32 – or to put it another way, 11.32 equals two TO WHAT POWER?

Luckily there is a mathematical function that answers this exact question: what power? Logarithms (or in this case… Dogarithms?) are a family of functions, and in this case I required log base 2 for the answer to my question: 11.32 equals two to what power? Unfortunately my phone calculator app doesn’t have a log base 2 button, but it does have a log button (meaning log base ten), and a well-know A-level formula tells me that:

#### $log_2 (11.32) = \frac{log (11.32)}{log (2)} = 3.50$

THE DOGARITHMIC ANSWER: Copper has doubled in weight (so presumably in size too) 3.50 times over 35 days (=5 weeks), so he doubles in size every $\frac{3.50}{35}=10$ days.

It’s always fun when higher level maths crops up in everyday situations! Thanks to the power of Dogarithms, this problem’s bark was worse than its bite.

WATCH OUT – COUNTER-INTUITIVE MATHS! If Copper doubles in length rather than in weight, then he becomes not just 2 times but eight times as big! This is because doubling in length most likely means he will also double in width and in height, so he becomes roughly  2x2x2=8 times as big! A similar line of reasoning tells us that a 46 inch television is four times as big as a 23 inch TV, not just twice as big.