How do you simplify ln 3 − 2 ln 8 + ln 16?

Mar 25, 2018

$\ln \left(\frac{3}{4}\right)$

Explanation:

We have to use the Logarithm Properties.

$\ln \left(3\right) - 2 \ln \left(8\right) + \ln \left(16\right)$

We can rewrite the initial expression using the Power rule in this way
$\ln \left(3\right) - \ln \left({8}^{2}\right) + \ln \left(16\right)$

$\ln \left(3\right) - \ln \left(64\right) + \ln \left(16\right)$

Here we use the Quotient rule
$\ln \left(\frac{3}{64}\right) + \ln \left(16\right)$

And here the Product rule
$\ln \left(\frac{3}{64} \cdot 16\right)$

$\ln \left(\frac{48}{64}\right)$

Finally we get the semplified version :
$\ln \left(\frac{3}{4}\right)$

Mar 25, 2018

$\ln \left(\frac{3}{4}\right)$

Explanation:

$\text{using the "color(blue)"laws of logarithms}$

•color(white)(x)logx^nhArrnlogx

•color(white)(x)logx+logyhArrlog(xy)

•color(white)(x)logx-logyhArrlog(x/y)

$\Rightarrow \ln 3 - \ln {8}^{2} + \ln 16$

$= \ln \left(\frac{3 \times 16}{64}\right) = \ln \left(\frac{48}{64}\right) = \ln \left(\frac{3}{4}\right)$