Solution

Simplify the following

$$e^{2\ln 3 - 3\ln 2}$$


The presence of the natural logs in the exponent should remind us of the inverse property that tells us $e^{\ln x} = x$. Unfortunately, we can't directly apply that here -- the exponent is a difference of multiples of natural logs, not a single natural log.

However, we can use the properties of logarithms to express the exponent in terms of a single natural log, which would then allow us to simplify the expression as described above.

To that end,

$$\begin{array}{rcll} e^{2\ln 3 - 3\ln 2} &=& e^{\ln 3^2 - \ln 2^3} & \scriptsize{\textrm{recalling } \log_b x^n = n \log_b x}\\\\ &=& e^{\ln 9 - \ln 8}\\\\ &=& e^{\ln (\frac{9}{8})} & \scriptsize{\textrm{recalling } \log_b x - \log_b y = \log_b \frac{x}{y}}\\\\ &=& \frac{9}{8} \end{array}$$