# Given log4=0.6021, log9=0.9542, and log12=1.-792, how do you find log 108,000?

Nov 21, 2016

$\log 108000 = 5.0334$

#### Explanation:

$\log 108000 = \log \left(108 \times 1000\right) = \log 108 + \log 1000$

= $3 + \log 108$

= $3 + \log \left(4 \times 27\right) = 3 + \log 4 + \log 27$

= $3 + \log 4 + \log {3}^{3}$

= $3 + \log 4 + 3 \log 3$

Now we have $\log 4 = 0.6021$ and as $\log {3}^{2} = \log 9 = 0.9542$,

we have $2 \log 3 = 0.9542$ and hence $\log 3 = \frac{0.9542}{2} = 0.4771$

Hence $\log 108000 = 3 + 0.6021 + 3 \times 0.4771$

= $3.6021 + 1.4313$

= $5.0334$