# How do you solve ln e^x -ln e^3 = ln e^9?

Jul 7, 2016

If you use the properties of logarithms, you will need to refer to these:

1. $\ln {a}^{b} = b \ln a$
2. $\ln e = 1$

OR

$\ln {e}^{z} = z$

where $a$ and $b$ are constants, $e = 2.718281828 \cdots$, and $z$ is either a constant or a variable. Therefore, we can do this:

$\ln {e}^{x} - \ln {e}^{3} = \ln {e}^{9}$

$\ln {e}^{x} = \ln {e}^{9} + \ln {e}^{3}$

$x \cancel{\ln e} = 9 \cancel{\ln e} + 3 \cancel{\ln e}$

$\textcolor{b l u e}{x} = 9 + 3 = \textcolor{b l u e}{12}$