How do you use a calculator to evaluate the expression log12 to four decimal places?

Dec 25, 2016

$\log 12 \approx 1.0792$

Explanation:

Since I remember (as is useful to remember):

$\log 2 \approx 0.30103$

$\log 3 \approx 0.47712125$

I can calculate:

$\log 12 = \log \left(2 \cdot 2 \cdot 3\right)$

$\textcolor{w h i t e}{\log 12} = \log 2 + \log 2 + \log 3$

$\textcolor{w h i t e}{\log 12} \approx 0.30103 + 0.30103 + 0.47712125$

$\textcolor{w h i t e}{\log 12} \approx 0.60206 + 0.47712125$

$\textcolor{w h i t e}{\log 12} \approx 1.07918125$

If you request $\log 12$ on a calculator, you will get a similar approximation.

If we truncated this to $4$ decimal places then we would get:

$1.0791$

but the following digit is $8 \ge 5$, so in order to round the value to $4$ decimal places we need to round up the final digit $1$ to $2$ to get:

$\log 12 \approx 1.0792$

$\textcolor{w h i t e}{}$
Footnote

If you remember good approximations for $\log 2$ and $\log 3$ then you can calculate approximations to $\log n$ for $n \in \left\{1 , 2 , 3 , 4 , 5 , 6 , 8 , 9 , 10 , 12 , 15 , 16 , 18 , 20 , \ldots\right\}$, i.e. any positive integer whose only prime factors are $2$, $3$ or $5$.

That seems to me to be good value for the sake of remembering a couple of numbers.