Rewrite expression using properties of logarithms. `ln(e^(5x)x^(3/2))`
First, apply the product property which is `ln (M*N)=ln M + lnN ` .
`=ln e^(5x) + ln x^(3/2)`
Next, apply the exponent property `ln M^a = alnM` .
`= 5xlne + 3/2lnx`
Note that `ln e` is the same as `log_e e` .
`= 5xlog_e e+3/2lnx`
And when the base of the logarithm and argument are the same it is equal to 1 `( log_b b= 1)` .
`=5x(1) + 3/2lnx`