Question

How can we express $\log \left(12\right)$ in terms of $\log \left(2\right)$ and $\log \left(3\right)$?

Answer 1

Solve the given logarithm functions

We know that factors of $12$ are $(2\times 2\times 3)$.

Taking $\log$ on both the sides, we get,

$\Rightarrow \log 12=\log \left(2\times 2\times 3\right)$
Now separating the term we get,
$\Rightarrow \log \left(12\right)=\log \left(2\right)+\log \left(2\right)+\log \left(3\right)\left[\because \log \left(abc\right)=\log \left(a\right)+\log \left(b\right)+\log \left(c\right)\right]$
Now, adding the like terms, we get,
$\Rightarrow \log \left(12\right)=2\log \left(2\right)+\log \left(3\right)$
Hence, we can express $\log \left(12\right)$ as $2\log \left(2\right)+\log \left(3\right)$