Natural Logarithm Laws

The laws for natural logarithms are the laws for logarithms written in base $e$:
\large \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.

\begin{align} \displaystyle \ln{4} + \ln{6} &= \ln{(4 \times 6)} \\ &= \ln{24} \end{align}

Example 2

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

\begin{align} \displaystyle \ln{15}-\ln{3} &= \ln{\dfrac{15}{3}} \\ &= \ln{5} \end{align}

Example 3

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

\begin{align} \displaystyle 2\ln{3} + 3\ln{2} &= \ln{3^2} + \ln{2^3} \\ &= \ln{9} + \ln{8} \\ &= \ln{(9 \times 8)} \\ &= \ln{72} \end{align}

Example 4

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

\begin{align} \displaystyle 4\ln{2} + 3 &= \ln{2^4} + 3 \times 1 \\ &= \ln{16} + 3\times \ln{e} \\ &= \ln{16} + \ln{e^3} \\ &= \ln{(16 \times e^3)} \\ &= \ln{(16e^3)} \end{align}

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