Question

Express log2(6!) in form a + log2(b) where a and b are integers and b is the smallest possible value?

Express `log_2(6!)` in the form `a + log_2(b)`, where a, b `in` z, and b is the smallest possible value

Answer 1

`log_2(6!)==4+log_2(45)`

Explanation:

`log_2(6!)=log_2(2*3*4*5*6)`
`=log_2(2*4)+log_2(3*5*2*3)`
`=log_2(2*4)+log_2(2)+log_2(3*5*3)`
`=log_2(2*4*2)+log_2(3*5*3)`
`=log_2(16)+log_2(45)`
`=cancel(log_2(2^4))^(color(red)(=4))+log_2(45)`
`=4+log_2(45)`

Because `gcd(45,2)=1`, `45` is the smallest `b` you can have.
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