Question

Write as a single logarithm.
ln(5) + 3/5 ln(32) − ln(4)

Answer 1

`ln(10)`

Explanation:

Let's rewrite the expression as a single logarithm using the next propierties:

`log(x*y)=log(x)+log(y)\\y*log(x)=log(x^(y) )\\log((x)/(y) )=log(x)-log(y)`

So:

`ln(5)+ln(32^{(3)/(5) } )-ln(4)`

Where:

`32^{(3)/(5) }=\sqrt[5]{32^(3) } =\sqrt[5]{32768} =8`

Therefore:

`ln(5)+ln(8)-ln(4)=ln(5*8)-ln(4)=ln((8*5)/(4))=ln((40)/(4))=ln(10)`