# How do you condense ln 3 − 2 ln 5 + ln 10?

Jun 28, 2016

$\ln \left(\frac{30}{25}\right)$

#### Explanation:

Adding logs means you were multiplying the numbers and dividing numbers means you were dividing the numbers.

$\ln 3 - 2 \ln 5 + \ln 10 \Rightarrow \ln 3 - \ln {5}^{2} + \ln 10$

=$\ln \left[\frac{3 \times 10}{5} ^ 2\right]$

=$\ln \left(\frac{30}{25}\right)$

Jun 28, 2016

$\ln \left(\frac{6}{5}\right)$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of logarithms}}$

•lnx+lny=ln(xy)........ (1)

•lnx-lny=ln(x/y)........ (2)

•lnx^nhArrnlnx........ (3)

These laws may be used on logarithms to any base.

Using (1) :$\ln 3 + \ln 10 = \ln \left(3 \times 10\right) = \ln 30$

Using (3): $2 \ln 5 = \ln {5}^{2} = \ln 25$

Using (2):$\ln 30 - \ln 25 = \ln \left(\frac{30}{25}\right) = \ln \left(\frac{6}{5}\right)$

$\Rightarrow \ln 3 - 2 \ln 5 + \ln 10 = \ln \left(\frac{6}{5}\right)$