# If log2= a and log3= b evaluate log(0.375)?

Mar 24, 2017

$\log 0.375 = b - 3 a$

#### Explanation:

Note that $\log 5 = \log \left(\frac{10}{2}\right) = \log 10 - \log 2 = \left(1 - a\right)$

We will now work on $\log 0.375$.

$\log 0.375$

= $\log \left(\frac{375}{1000}\right)$

= $\log 375 - \log 1000$

= $\log \left(3 \times 5 \times 5 \times 5\right) - 3$

= $\log 3 + 3 \log 5 - 3$

= $b + 3 \left(1 - a\right) - 3$

= $b + 3 - 3 a - 3$

= $b - 3 a$

Alternatively , one can also write

$\log 0.375 = \log \left(\frac{3}{8}\right) = \log 3 - \log 8 = \log 3 - \log {2}^{3}$

= $\log 3 - 3 \log 2 = b - 3 a$