How do you condense 4 ln 2 - ln 3 - 1?

Feb 25, 2017

$4 \ln 2 - \ln 3 - 1 = \ln \left(\frac{16}{3 e}\right)$

Explanation:

$4 \ln 2 - \ln 3 - 1$

We know that: $a \ln b = \ln {b}^{a}$, therefore,

$= \ln {2}^{4} - \ln 3 - 1$
$= \ln 16 - \ln 3 - 1$

We know that: $\ln a - \ln b = \ln \left(a \cdot \frac{1}{b}\right)$, therefore,

$= \ln \left(16 \cdot \frac{1}{3}\right) - 1$
$= \ln \left(\frac{16}{3}\right) - 1$

We know that: $\ln e = 1$, therefore,

$= \ln \left(\frac{16}{3}\right) - \ln e$
$= \ln \left(\frac{16}{3} \cdot \frac{1}{e}\right)$
$= \ln \left(\frac{16}{3 e}\right)$

Hence, $4 \ln 2 - \ln 3 - 1 = \ln \left(\frac{16}{3 e}\right)$