Natural Logarithm Laws

The laws for natural logarithms are the laws for logarithms written in base $e$: $$ \begin{align} \displaystyle \ln{x} + \ln{y} &= \ln{(xy)} \\ \ln{x} - \ln{y} &= \ln{\dfrac{x}{y}} \\ \ln{x^n} &= n\ln{x} \\ \ln{e} &= 1 \\ \end{align} $$ Note that $\ln{x}=\log_{e}{x}$ and $x>0,y>0$.

Example 1

Use the laws of logarithms to write $\ln{4} + \ln{6}$ as a single logarithm.

Example 2

Use the laws of logarithms to write $\ln{15} - \ln{3}$ as a single logarithm.

Example 3

Use the laws of logarithms to write $2\ln{3} + 3\ln{2}$ as a single logarithm.

Example 4

Use the laws of logarithms to write $4\ln{2} + 3$ as a single logarithm.