Secrets of the mathematical ninja: Estimating natural logs

Natural logs are just about the easiest part of the A-level syllabus to look like a god in — because they’re wrongly seen as difficult. In fact, once you know a handful of tricks, you can rattle off things such as ‘log of 12 is about 2.5, of course…’ (it’s 2.485) with confidence and aplomb.

Here are a few key values to know by heart:

• $\ln(1) = 0$ (exact)
• $\ln(2) = 0.7$ (-1%)
• $\ln(3) = 1.1$ (-0.1%)
• $\ln(5) = 1.6$ (+0.6%)

And a few rules you need to know whether you want to be a ninja or not:

• $\ln(ab) = \ln(a) + \ln(b)$
• $\ln(a^b) = b \ln(a)$
• $\ln(\frac{a}{b}) = \ln(a) – \ln(b)$

So, if you want to know $\ln(100)$, you can say ‘that’s $2 \ln (10)$’, and then ‘$\ln(10)$ is $\ln(2) + \ln(5)$, which is $0.7 + 1.6$, about $2.3$ — so $\ln(100)$ is about 4.6′. (It’s 4.605).

There’s also a really helpful approximation for natural logs of numbers near 1:

$$\ln(1 + x) \approx x$$

That goes deeper than it looks. If you’re looking for the log of a number that’s slightly off something you know — say you want to know $\ln(17)$ for some reason, and you spot straightaway that $\ln(16)$ is $4 \ln(2)$, or 2.8 (less 1%, which is 2.77). You can say ‘I’d need to add about 6% (of 16) on to 16 to get 17 — so I need to add 0.06 on to $\ln(16)$ to get $\ln(17)$, which will make it 2.83. (It’s 2.833).

But wait — there’s more. If you’re smart with your dividing, you can solve the silly powers questions they give you in C2, such as $5^x = 17$.

When you rearrange that, you get $x = \log_5 (17)$, which is the same as $\frac{\ln(17)}{\ln(5)}$. You know both of those: it’s $\frac{2.83}{1.6}$. Rounding that off to $\frac{2.8}{1.6}$ or $\frac{28}{16}$, you get an answer of $\frac{7}{4}$, which is 1.75. (The actual answer is 1.76 — which isn’t at all shabby without a calculator).

(There’s a serious point to this: if you can work this natural logs stuff out in your head, you can look at your calculator and say ‘that looks ok’ or ‘that looks dubious’. Looking ok isn’t a guarantee that you have the right answer, but looking dubious is a good sign that you need to check your work more carefully and figure out whether your mistake is in the estimate or the sum.)

Colin

Colin is a maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives in Dorset with an espresso pot and nothing to prove.