# Rewrite expression using properties of logarithms. `ln(e^(5x)x^(3/2))`

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`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`

`=5x+3/2lnx`

**Hence,**

**`ln(e^(5x)*x^(3/2))=5x+3/2lnx` .**