Given log4=0.6021, log9=0.9542, and log12=1.-792, how do you find log 36?

Dec 3, 2016

Use one of the laws of logarithms: $\log \left(a \cdot b\right) = \log a + \log b .$
$\log 36 = 1.5563 .$

Explanation:

We want to know $\log 36$, but we don't have a value for it. But we do have values for $\log 4$, $\log 9$, and $\log 12$. Can we use the above law of logarithms to find something equal to $\log 36$, in terms of the $\log$ values we're given?

Yes, we can! Because $36 = 4 \cdot 9$, we can write:

$\log 36 = \log \left(4 \cdot 9\right)$
$\textcolor{w h i t e}{\log 36} = \log 4 + \log 9$
$\textcolor{w h i t e}{\log 36} = 0.6021 + 0.9542$
$\textcolor{w h i t e}{\log 36} = 1.5563$

So, based on the values given for $\log 4$ and $\log 9$, we have:
$\log 36 = 1.5563 .$

Bonus:

What this log rule states is that multiplying two numbers is equivalent to adding their logarithms. So, if you were asked to multiply two big numbers together very quickly, you could convert this to simple addition by converting the numbers to their logs with a log table, adding the logs, and converting the sum back using the same log table.

If you're interested, look up "log tables" and "slide rules". This is how multiplication was done in the 1950s and 60s, before pocket calculators were around.